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1. Shard Calculation

Query:

Response:

Calculated Shard: 115 (from laksa198)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa115.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.probabilitycourse.com/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.probabilitycourse.com/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.probabilitycourse.com\/","crawl_time":1776465423,"first_indexed_time":1469553798,"http_code":200,"src_unparsed":"com,probabilitycourse!www,\/ s443","src_root_hash":"8531609719846996715","history_drop_reason":null,"meta_title":"Probability, Statistics & Random Processes | Free Textbook | Course","meta_descriptions":[],"attrs_boilerpipe_text":"News:\nOnline Spring 2026 Courses:\nECE 150 - Making Better Decisions by Humans and AI\n(4 credits)\nECE 214 - Probability and Statistics\n(4 credits at UMass Amherst)\nECE 579 - Math Tools for Data Science & Machine Learning\n(3 credits)\nECE 603 - Probability & Random Process\n(3 credits)\nNew book release:\nPractical uncertainty:\nUseful Ideas in Decision-Making, Risk, Randomness, & AI\nis now available on Amazon in\nprint\n,\nKindle\n, and\naudiobook\nformats.\nInterested readers can visit our\nwebsite\nto download a free sample.\nStep-by-Step Solutions to Selected Problems in Signals & Systems\nby Hamid Saeedi, Hossein Pishro-Nik\nThis book is available on\nAmazon\n.\nInterested students can view\nsample problems\nand\nsample solutions\n.\nA chapter on\nsimulation using Python\nwas added!\nWelcome\nThis site is the homepage of the textbook Introduction to Probability, \n\t\t\t\tStatistics, and Random Processes by Hossein Pishro-Nik. It is an open access \n\t\t\t\tpeer-reviewed textbook intended for undergraduate as well as first-year \n\t\t\t\tgraduate level courses on the subject. This probability textbook can be used by both students and \n\t\t\t\tpractitioners in engineering, mathematics, finance, and other related fields.\nThe site includes:\nThe entire textbook\nShort video lectures that aid in learning the material\nOnline probability calculators for important functions and distributions\nA solutions manual for instructors\nLecture slides\nThe print version of the book is available through Amazon\nhere\n.\nBook Coverage\nThis probability and statistics textbook covers:\nBasic concepts such as random experiments, probability axioms,\n\t\t\t\t\tconditional probability, and counting methods\nSingle and multiple random variables (discrete, continuous, and\n\t\t\t\t\tmixed), as well as moment-generating functions, characteristic \n\t\t\t\t\tfunctions, random vectors, and inequalities\nLimit theorems and convergence\nIntroduction to mathematical statistics, in particular, Bayesian and classical statistics\nRandom processes including processing of random signals, Poisson\n\t\t\t\t\tprocesses, discrete-time and continuous-time Markov chains, and \n\t\t\t\t\tBrownian motion\nSimulation using MATLAB, R, and Python\nHow to cite\nYou can cite this textbook as:\nH. Pishro-Nik, \"Introduction to probability, statistics, and random processes\", available at\nhttps:\/\/www.probabilitycourse.com\n, Kappa Research LLC, 2014.\nStudent’s Solutions Guide\nSince the textbook's initial publication, many requested the distribution \n\t\t\tof solutions to the problems in the textbook. We published the student’s \n\t\t\tsolutions guide which includes guided solutions to the\nodd-numbered\nend-of-chapter problems.\nThis guide is available on Amazon in both\nprint\nand\nkindle\nformat:\nAbout the Author\nH. Pishro-Nik\nis a professor in the Department of Electrical and \n\t\t\t\tComputer Engineering at the University of Massachusetts Amherst. He \n\t\t\t\treceived his B.S. degree from Sharif University of Technology, and M.Sc. and \n\t\t\t\tPh.D. degrees from Georgia Institute of Technology, all in Electrical and \n\t\t\t\tComputer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making.","attrs_markdown":"- [HOME](https:\/\/www.probabilitycourse.com\/)\n- [VIDEOS](https:\/\/www.probabilitycourse.com\/videos\/videos.php)\n- [CALCULATOR](https:\/\/www.probabilitycourse.com\/calculator\/calculator.php)\n- [COMMENTS](https:\/\/www.probabilitycourse.com\/comments.php)\n- [COURSES](https:\/\/www.probabilitycourse.com\/courses.php)\n- [FOR INSTRUCTOR](https:\/\/www.probabilitycourse.com\/for_instructors.php)\n- [LOG IN](https:\/\/www.probabilitycourse.com\/Login\/mobile_login.php)\n\n[![](https:\/\/www.probabilitycourse.com\/images\/Template\/logo_probability_website.png)](https:\/\/www.probabilitycourse.com\/)\n\nChapters\n\nMenu\n\n- [HOME](https:\/\/www.probabilitycourse.com\/)\n- [VIDEOS](https:\/\/www.probabilitycourse.com\/videos\/videos.php)\n- [CALCULATOR](https:\/\/www.probabilitycourse.com\/calculator\/calculator.php)\n- [COMMENTS](https:\/\/www.probabilitycourse.com\/comments.php)\n- [COURSES](https:\/\/www.probabilitycourse.com\/courses.php)\n- [FOR INSTRUCTORS](https:\/\/www.probabilitycourse.com\/for_instructors.php)\n\n- [Sign In]()\n\n[Forgot password?](https:\/\/www.probabilitycourse.com\/Login\/forgot_password.php)\n\nOpen Menu\n\n- [0 Preface](https:\/\/www.probabilitycourse.com\/preface.php)\n- [1 Basic Concepts]()\n  - [1\\.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter1\/1_0_0_introduction.php)\n  - [1\\.1 Introduction]()\n    - [1\\.1.0 What Is Probability?](https:\/\/www.probabilitycourse.com\/chapter1\/1_1_0_what_is_probability.php)\n    - [1\\.1.1 Example](https:\/\/www.probabilitycourse.com\/chapter1\/1_1_1_example.php)\n  - [1\\.2 Review of Set Theory]()\n    - [1\\.2.0 Review](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_0_review_set_theory.php)\n    - [1\\.2.1 Venn Diagrams](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_1_venn.php)\n    - [1\\.2.2 Set Operations](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_2_set_operations.php)\n    - [1\\.2.3 Cardinality](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_3_cardinality.php)\n    - [1\\.2.4 Functions](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_4_functions.php)\n    - [1\\.2.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter1\/1_2_5_solved1.php)\n  - [1\\.3 Random Experiments and Probabilities]()\n    - [1\\.3.1 Random Experiments](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_1_random_experiments.php)\n    - [1\\.3.2 Probability](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_2_probability.php)\n    - [1\\.3.3 Finding Probabilities](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_3_finding_probabilities.php)\n    - [1\\.3.4 Discrete Models](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_4_discrete_models.php)\n    - [1\\.3.5 Continuous Models](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_5_continuous_models.php)\n    - [1\\.3.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter1\/1_3_6_solved2.php)\n  - [1\\.4 Conditional Probability]()\n    - [1\\.4.0 Conditional Probability](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_0_conditional_probability.php)\n    - [1\\.4.1 Independence](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_1_independence.php)\n    - [1\\.4.2 Law of Total Probability](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_2_total_probability.php)\n    - [1\\.4.3 Bayes' Rule](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_3_bayes_rule.php)\n    - [1\\.4.4 Conditional Independence](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_4_conditional_independence.php)\n    - [1\\.4.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter1\/1_4_5_solved3.php)\n  - [1\\.5 Problems]()\n    - [1\\.5.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter1\/1_5_0_chapter1_problems.php)\n- [2 Combinatorics: Counting Methods]()\n  - [2\\.1 Combinatorics]()\n    - [2\\.1.0 Finding Probabilities with Counting Methods](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_0_counting.php)\n    - [2\\.1.1 Ordered with Replacement](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_1_ordered_with_replacement.php)\n    - [2\\.1.2 Ordered without Replacement](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_2_ordered_without_replacement.php)\n    - [2\\.1.3 Unordered without Replacement](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_3_unordered_without_replacement.php)\n    - [2\\.1.4 Unordered with Replacement](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_4_unordered_with_replacement.php)\n    - [2\\.1.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter2\/2_1_5_solved2_1.php)\n  - [2\\.2 Problems]()\n    - [2\\.2.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter2\/2_3_0_chapter2_problems.php)\n- [3 Discrete Random Variables]()\n  - [3\\.1 Basic Concepts]()\n    - [3\\.1.1 Random Variables](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_1_random_variables.php)\n    - [3\\.1.2 Discrete Random Variables](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_2_discrete_random_var.php)\n    - [3\\.1.3 Probability Mass Function](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_3_pmf.php)\n    - [3\\.1.4 Independent Random Variables](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_4_independent_random_var.php)\n    - [3\\.1.5 Special Distributions](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_5_special_discrete_distr.php)\n    - [3\\.1.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter3\/3_1_6_solved3_1.php)\n  - [3\\.2 More about Discrete Random Variables]()\n    - [3\\.2.1 Cumulative Distribution Function](https:\/\/www.probabilitycourse.com\/chapter3\/3_2_1_cdf.php)\n    - [3\\.2.2 Expectation](https:\/\/www.probabilitycourse.com\/chapter3\/3_2_2_expectation.php)\n    - [3\\.2.3 Functions of Random Variables](https:\/\/www.probabilitycourse.com\/chapter3\/3_2_3_functions_random_var.php)\n    - [3\\.2.4 Variance](https:\/\/www.probabilitycourse.com\/chapter3\/3_2_4_variance.php)\n    - [3\\.2.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter3\/3_2_5_solved3_2.php)\n  - [3\\.3 Problems]()\n    - [3\\.3.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter3\/3_3_0_chapter3_problems.php)\n- [4 Continuous and Mixed Random Variables]()\n  - [4\\.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter4\/4_0_0_intro.php)\n  - [4\\.1 Continuous Random Variables]()\n    - [4\\.1.0 Continuous Random Variables and their Distributions](https:\/\/www.probabilitycourse.com\/chapter4\/4_1_0_continuous_random_vars_distributions.php)\n    - [4\\.1.1 Probability Density Function](https:\/\/www.probabilitycourse.com\/chapter4\/4_1_1_pdf.php)\n    - [4\\.1.2 Expected Value and Variance](https:\/\/www.probabilitycourse.com\/chapter4\/4_1_2_expected_val_variance.php)\n    - [4\\.1.3 Functions of Continuous Random Variables](https:\/\/www.probabilitycourse.com\/chapter4\/4_1_3_functions_continuous_var.php)\n    - [4\\.1.4 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter4\/4_1_4_solved4_1.php)\n  - [4\\.2 Special Distributions]()\n    - [4\\.2.1 Uniform Distribution](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_1_uniform.php)\n    - [4\\.2.2 Exponential Distribution](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_2_exponential.php)\n    - [4\\.2.3 Normal (Gaussian) Distribution](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_3_normal.php)\n    - [4\\.2.4 Gamma Distribution](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_4_Gamma_distribution.php)\n    - [4\\.2.5 Other Distributions](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_5_other_distr.php)\n    - [4\\.2.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter4\/4_2_6_solved4_2.php)\n  - [4\\.3 Mixed Random Variables]()\n    - [4\\.3.1 Mixed Random Variables](https:\/\/www.probabilitycourse.com\/chapter4\/4_3_1_mixed.php)\n    - [4\\.3.2 Using the Delta Function](https:\/\/www.probabilitycourse.com\/chapter4\/4_3_2_delta_function.php)\n    - [4\\.3.3 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter4\/4_3_3_solved4_3.php)\n  - [4\\.4 Problems]()\n    - [4\\.4.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter4\/4_4_0_chapter4_problems.php)\n- [5 Joint Distributions]()\n  - [5\\.1 Two Discrete Random Variables]()\n    - [5\\.1.0 Two Random Variables](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_0_joint_distributions.php)\n    - [5\\.1.1 Joint Probability Mass Function (PMF)](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_1_joint_pmf.php)\n    - [5\\.1.2 Joint Cumulative Distribution Function (CDF)](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_2_joint_cdf.php)\n    - [5\\.1.3 Conditioning and Independence](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_3_conditioning_independence.php)\n    - [5\\.1.4 Functions of Two Random Variables](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_4_functions_two_variables.php)\n    - [5\\.1.5 Conditional Expectation](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_5_conditional_expectation.php)\n    - [5\\.1.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter5\/5_1_6_solved_prob.php)\n  - [5\\.2 Two Continuous Random Variables]()\n    - [5\\.2.0 Two Continuous Random Variables](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_0_continuous_vars.php)\n    - [5\\.2.1 Joint Probability Density Function](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_1_joint_pdf.php)\n    - [5\\.2.2 Joint Cumulative Distribution Function](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_2_joint_cdf.php)\n    - [5\\.2.3 Conditioning and Independence](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_3_conditioning_independence.php)\n    - [5\\.2.4 Functions of Two Continuous Random Variables](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_4_functions.php)\n    - [5\\.2.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter5\/5_2_5_solved_prob.php)\n  - [5\\.3 More Topics]()\n    - [5\\.3.1 Covariance and Correlation](https:\/\/www.probabilitycourse.com\/chapter5\/5_3_1_covariance_correlation.php)\n    - [5\\.3.2 Bivariate Normal Distribution](https:\/\/www.probabilitycourse.com\/chapter5\/5_3_2_bivariate_normal_dist.php)\n    - [5\\.3.3 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter5\/5_3_3_solved_probs.php)\n  - [5\\.4 Problems]()\n    - [5\\.4.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter5\/5_4_0_chapter_problems.php)\n- [6 Multiple Random Variables]()\n  - [6\\.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter6\/6_0_0_intro.php)\n  - [6\\.1 Methods for More Than Two Random Variables]()\n    - [6\\.1.1 Joint Distributions and Independence](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_1_joint_distributions_independence.php)\n    - [6\\.1.2 Sums of Random Variables](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_2_sums_random_variables.php)\n    - [6\\.1.3 Moment Generating Functions](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_3_moment_functions.php)\n    - [6\\.1.4 Characteristic Functions](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_4_characteristic_functions.php)\n    - [6\\.1.5 Random Vectors](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_5_random_vectors.php)\n    - [6\\.1.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter6\/6_1_6_solved_probs.php)\n  - [6\\.2 Probability Bounds]()\n    - [6\\.2.0 Probability Bounds](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_0_probability_bounds.php)\n    - [6\\.2.1 Union Bound and Extension](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_1_union_bound_and_exten.php)\n    - [6\\.2.2 Markov Chebyshev Inequalities](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_2_markov_chebyshev_inequalities.php)\n    - [6\\.2.3 Chernoff Bounds](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_3_chernoff_bounds.php)\n    - [6\\.2.4 Cauchy Schwarz Inequality](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_4_cauchy_schwarz.php)\n    - [6\\.2.5 Jensen's Inequality](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_5_jensen's_inequality.php)\n    - [6\\.2.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter6\/6_2_6_solved6_2.php)\n  - [6\\.3 Problems]()\n    - [6\\.3.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter6\/6_3_0_chapter_problems.php)\n- [7 Limit Theorems and Convergence of Random Variables]()\n  - [7\\.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter7\/7_0_0_intro.php)\n  - [7\\.1 Limit Theorems]()\n    - [7\\.1.0 Limit Theorems](https:\/\/www.probabilitycourse.com\/chapter7\/7_1_0_limit_theorems.php)\n    - [7\\.1.1 Law of Large Numbers](https:\/\/www.probabilitycourse.com\/chapter7\/7_1_1_law_of_large_numbers.php)\n    - [7\\.1.2 Central Limit Theorem (CLT)](https:\/\/www.probabilitycourse.com\/chapter7\/7_1_2_central_limit_theorem.php)\n    - [7\\.1.3 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter7\/7_1_3_solved_probs.php)\n  - [7\\.2 Convergence of Random Variables]()\n    - [7\\.2.0 Convergence of Random Variables](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_0_convergence_of_random_variables.php)\n    - [7\\.2.1 Convergence of Sequence of Numbers](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_1_convergence_of_a_seq_of_nums.php)\n    - [7\\.2.2 Sequence of Random Variables](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_2_sequence_of_random_variables.php)\n    - [7\\.2.3 Different Types of Convergence for Sequences of Random Variables](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_3_different_types_of_convergence_for_sequences_of_random_variables.php)\n    - [7\\.2.4 Convergence in Distribution](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_4_convergence_in_distribution.php)\n    - [7\\.2.5 Convergence in Probability](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_5_convergence_in_probability.php)\n    - [7\\.2.6 Convergence in Mean](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_6_convergence_in_mean.php)\n    - [7\\.2.7 Almost Sure Convergence](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_7_almost_sure_convergence.php)\n    - [7\\.2.8 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter7\/7_2_8_solved_probs.php)\n  - [7\\.3 Problems]()\n    - [7\\.3.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter7\/7_3_0_chapter_problem.php)\n- [8 Statistical Inference I: Classical Methods]()\n  - [8\\.1 Introduction]()\n    - [8\\.1.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter8\/8_1_0_intro.php)\n    - [8\\.1.1 Random Sampling](https:\/\/www.probabilitycourse.com\/chapter8\/8_1_1_random_sampling.php)\n  - [8\\.2 Point Estimation]()\n    - [8\\.2.0 Point Estimation](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_0_point_estimation.php)\n    - [8\\.2.1 Evaluating Estimators](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_1_evaluating_estimators.php)\n    - [8\\.2.2 Point Estimators for Mean and Variance](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_2_point_estimators_for_mean_and_var.php)\n    - [8\\.2.3 Maximum Likelihood Estimation (MLE)](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_3_max_likelihood_estimation.php)\n    - [8\\.2.4 Asymptotic Properties of MLEs](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_4_asymptotic_probs_of_MLE.php)\n    - [8\\.2.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter8\/8_2_5_solved_probs.php)\n  - [8\\.3 Interval Estimation (Confidence Intervals)]()\n    - [8\\.3.0 Interval Estimation (Confidence Intervals)](https:\/\/www.probabilitycourse.com\/chapter8\/8_3_0_interval_estimation.php)\n    - [8\\.3.1 The general framework of Interval Estimation](https:\/\/www.probabilitycourse.com\/chapter8\/8_3_1_gen_framework_of_int_estimation.php)\n    - [8\\.3.2 Finding Interval Estimators](https:\/\/www.probabilitycourse.com\/chapter8\/8_3_2_finding_interval_estimators.php)\n    - [8\\.3.3 Confidence Intervals for Normal Samples](https:\/\/www.probabilitycourse.com\/chapter8\/8_3_3_confidence_intervals_for_norm_samples.php)\n    - [8\\.3.4 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter8\/8_3_4_solved_probs.php)\n  - [8\\.4 Hypothesis Testing]()\n    - [8\\.4.1 Introduction](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_1_intro.php)\n    - [8\\.4.2 General Setting and Definitions](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_2_general_setting_definitions.php)\n    - [8\\.4.3 Hypothesis Testing for the Mean](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_3_hypothesis_testing_for_mean.php)\n    - [8\\.4.4 P-Values](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_4_p_vals.php)\n    - [8\\.4.5 Likelihood Ratio Tests](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_5_likelihood_ratio_tests.php)\n    - [8\\.4.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter8\/8_4_6_solved_probs.php)\n  - [8\\.5 Linear Regression]()\n    - [8\\.5.0 Linear Regression](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_0_linear_regression.php)\n    - [8\\.5.1 Simple Linear Regression Model](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_1_simple_linear_regression_model.php)\n    - [8\\.5.2 The First Method for Finding beta](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_2_first_method_for_finding_beta.php)\n    - [8\\.5.3 The Method of Least Squares](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_3_the_method_of_least_squares.php)\n    - [8\\.5.4 Extensions and Issues](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_4_extensions_and_issues.php)\n    - [8\\.5.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter8\/8_5_5_solved_probs.php)\n  - [8\\.6 Problems]()\n    - [8\\.6.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter8\/8_6_0_ch_probs.php)\n- [9 Statistical Inference II: Bayesian Inference]()\n  - [9\\.1 Bayesian Inference]()\n    - [9\\.1.0 Bayesian Inference](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_0_bayesian_inference.php)\n    - [9\\.1.1 Prior and Posterior](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_1_prior_and_posterior.php)\n    - [9\\.1.2 Maximum A Posteriori (MAP) Estimation](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_2_MAP_estimation.php)\n    - [9\\.1.3 Comparison to ML Estimation](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_3_comparison_to_ML_estimation.php)\n    - [9\\.1.4 Conditional Expectation (MMSE)](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_4_conditional_expectation_MMSE.php)\n    - [9\\.1.5 Mean Squared Error (MSE)](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_5_mean_squared_error_MSE.php)\n    - [9\\.1.6 Linear MMSE Estimation of Random Variables](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_6_linear_MMSE_estimat_of_random_vars.php)\n    - [9\\.1.7 Estimation for Random Vectors](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_7_estimation_for_random_vectors.php)\n    - [9\\.1.8 Bayesian Hypothesis Testing](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_8_bayesian_hypothesis_testing.php)\n    - [9\\.1.9 Bayesian Interval Estimation](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_9_bayesian_interval_estimation.php)\n    - [9\\.1.10 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter9\/9_1_10_solved_probs.php)\n  - [9\\.2 Problems]()\n    - [9\\.2.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter9\/9_2_0_ch_probs.php)\n- [10 Introduction to Random Processes]()\n  - [10\\.1 Basic Concepts]()\n    - [10\\.1.0 Basic Concepts](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_0_basic_concepts.php)\n    - [10\\.1.1 PDFs and CDFs](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_1_PDFs_and_CDFs.php)\n    - [10\\.1.2 Mean and Correlation Functions](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_2_mean_and_correlation_functions.php)\n    - [10\\.1.3 Multiple Random Processes](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_3_multiple_random_processes.php)\n    - [10\\.1.4 Stationary Processes](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_4_stationary_processes.php)\n    - [10\\.1.5 Gaussian Random Processes](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_5_gaussian_random_processes.php)\n    - [10\\.1.6 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter10\/10_1_6_solved_probs.php)\n  - [10\\.2 Processing of Random Signals]()\n    - [10\\.2.0 Processing of Random Signals](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_0_processing_of_random_signals.php)\n    - [10\\.2.1 Power Spectral Density](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_1_power_spectral_density.php)\n    - [10\\.2.2 Linear Time-Invariant (LTI) Systems with Random Inputs](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_2_LTI_systems_with_random_inputs.php)\n    - [10\\.2.3 Power in a Frequency Band](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_3_power_in_a_frequency_band.php)\n    - [10\\.2.4 White Noise](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_4_white_noise.php)\n    - [10\\.2.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter10\/10_2_5_solved_probs.php)\n  - [10\\.3 Problems]()\n    - [10\\.3.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter10\/10_3_0_ch_probs.php)\n- [11 Some Important Random Processes]()\n  - [11\\.1 Poisson Processes]()\n    - [11\\.1.0 Introduction](https:\/\/www.probabilitycourse.com\/chapter11\/11_0_0_intro.php)\n    - [11\\.1.1 Counting Processes](https:\/\/www.probabilitycourse.com\/chapter11\/11_1_1_counting_processes.php)\n    - [11\\.1.2 Basic Concepts of the Poisson Process](https:\/\/www.probabilitycourse.com\/chapter11\/11_1_2_basic_concepts_of_the_poisson_process.php)\n    - [11\\.1.3 Merging and Splitting Poisson Processes](https:\/\/www.probabilitycourse.com\/chapter11\/11_1_3_merging_and_splitting_poisson_processes.php)\n    - [11\\.1.4 Nonhomogeneous Poisson Processes](https:\/\/www.probabilitycourse.com\/chapter11\/11_1_4_nonhomogeneous_poisson_processes.php)\n    - [11\\.1.5 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter11\/11_1_5_solved_probs.php)\n  - [11\\.2 Discrete-Time Markov Chains]()\n    - [11\\.2.1 Introduction](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_1_introduction.php)\n    - [11\\.2.2 State Transition Matrix and Diagram](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_2_state_transition_matrix_and_diagram.php)\n    - [11\\.2.3 Probability Distributions](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_3_probability_distributions.php)\n    - [11\\.2.4 Classification of States](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_4_classification_of_states.php)\n    - [11\\.2.5 Using the Law of Total Probability with Recursion](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_5_using_the_law_of_total_probability_with_recursion.php)\n    - [11\\.2.6 Stationary and Limiting Distributions](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_6_stationary_and_limiting_distributions.php)\n    - [11\\.2.7 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter11\/11_2_7_solved_probs.php)\n  - [11\\.3 Continuous-Time Markov Chains]()\n    - [11\\.3.1 Introduction](https:\/\/www.probabilitycourse.com\/chapter11\/11_3_1_introduction.php)\n    - [11\\.3.2 Stationary and Limiting Distributions](https:\/\/www.probabilitycourse.com\/chapter11\/11_3_2_stationary_and_limiting_distributions.php)\n    - [11\\.3.3 The Generator Matrix](https:\/\/www.probabilitycourse.com\/chapter11\/11_3_3_the_generator_matrix.php)\n    - [11\\.3.4 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter11\/11_3_4_solved_probs.php)\n  - [11\\.4 Brownian Motion (Wiener Process)]()\n    - [11\\.4.0 Brownian Motion (Wiener Process)](https:\/\/www.probabilitycourse.com\/chapter11\/11_4_0_brownian_motion_wiener_process.php)\n    - [11\\.4.1 Brownian Motion as the Limit of a Symmetric Random Walk](https:\/\/www.probabilitycourse.com\/chapter11\/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php)\n    - [1\\.4.2 Definition and Some Properties](https:\/\/www.probabilitycourse.com\/chapter11\/11_4_2_definition_and_some_properties.php)\n    - [11\\.4.3 Solved Problems](https:\/\/www.probabilitycourse.com\/chapter11\/11_4_3_solved_probs.php)\n  - [11\\.5 Problems]()\n    - [11\\.5.0 End of Chapter Problems](https:\/\/www.probabilitycourse.com\/chapter11\/11_5_0_end_of_chapter_problems.php)\n- [12 Introduction to Simulation Using MATLAB](https:\/\/www.probabilitycourse.com\/chapter12\/chapter12.php)\n- [13 Introduction to Simulation Using R](https:\/\/www.probabilitycourse.com\/chapter13\/chapter13.php)\n- [14 Introduction to Simulation Using Python](https:\/\/www.probabilitycourse.com\/chapter14\/chapter14.php)\n- [15 Recursive Methods](https:\/\/www.probabilitycourse.com\/chapter15\/chapter15.php)\n- [Appendix]()\n  - [Some Important Distributions](https:\/\/www.probabilitycourse.com\/appendix\/some_important_distributions.php)\n  - [Review of the Fourier Transform](https:\/\/www.probabilitycourse.com\/appendix\/review_fourier_transform.php)\n- [Bibliography](https:\/\/www.probabilitycourse.com\/bibliography.php)\n\n## News:\n- **Online Spring 2026 Courses:**\n  - [ECE 150 - Making Better Decisions by Humans and AI](https:\/\/www.probabilitycourse.com\/courses.php#ECE150) (4 credits)\n  - [ECE 214 - Probability and Statistics](https:\/\/www.probabilitycourse.com\/courses.php#ECE214) (4 credits at UMass Amherst)\n  - [ECE 579 - Math Tools for Data Science & Machine Learning](https:\/\/www.probabilitycourse.com\/courses.php#ECE579) (3 credits)\n  - [ECE 603 - Probability & Random Process](https:\/\/www.probabilitycourse.com\/courses.php#ECE603) (3 credits)\n- **A chapter on [simulation using Python](https:\/\/www.probabilitycourse.com\/chapter14\/chapter14.php) was added\\!**\n***\n# Welcome\nThis site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. This probability textbook can be used by both students and practitioners in engineering, mathematics, finance, and other related fields.\n\nThe site includes:\n\n- The entire textbook\n- Short video lectures that aid in learning the material\n- Online probability calculators for important functions and distributions\n- A solutions manual for instructors\n- [Lecture slides](https:\/\/www.dropbox.com\/sh\/iuhdrbq1yqf8t8q\/AAAV6npl5xFkqfvTd2jSEjjwa?dl=0)\n\nThe print version of the book is available through Amazon [here](https:\/\/www.amazon.com\/Introduction-Probability-Statistics-Random-Processes\/dp\/0990637204\/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik).\n\n[![Book Cover](https:\/\/www.probabilitycourse.com\/images\/home\/book_cover.jpg)](https:\/\/www.amazon.com\/Introduction-Probability-Statistics-Random-Processes\/dp\/0990637204\/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik)\n\n# Book Coverage\nThis probability and statistics textbook covers:\n\n- Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods\n- Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities\n- Limit theorems and convergence\n- Introduction to mathematical statistics, in particular, Bayesian and classical statistics\n- Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion\n- Simulation using MATLAB, R, and Python\n# How to cite\nYou can cite this textbook as:\n\nH. Pishro-Nik, \"Introduction to probability, statistics, and random processes\", available at <https:\/\/www.probabilitycourse.com>, Kappa Research LLC, 2014.\n\n# Student’s Solutions Guide\nSince the textbook's initial publication, many requested the distribution of solutions to the problems in the textbook. We published the student’s solutions guide which includes guided solutions to the **odd-numbered** end-of-chapter problems.\n\nThis guide is available on Amazon in both [print](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes\/dp\/0990637212\/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) and [kindle](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes-ebook\/dp\/B01HBRRHLY\/ref=sr_1_3?ie=UTF8&qid=1466517931&sr=8-3&keywords=pishro-nik) format:\n\n[![Solution Book Cover](https:\/\/www.probabilitycourse.com\/images\/home\/solution_cover.png)](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes\/dp\/0990637212\/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik)\n\n# About the Author\n[H. Pishro-Nik](https:\/\/websites.umass.edu\/pishro\/) is a professor in the Department of Electrical and Computer Engineering at the University of Massachusetts Amherst. He received his B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from Georgia Institute of Technology, all in Electrical and Computer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making.\n\n[![Creative Commons License](https:\/\/i.creativecommons.org\/l\/by-nc-nd\/3.0\/88x31.png)](https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/deed.en_US)  \nIntroduction to Probability by  [Hossein Pishro-Nik](https:\/\/websites.umass.edu\/pishro\/) is licensed under a [Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License](https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/deed.en_US)","attrs_readable_markdown":"## News:\n- **Online Spring 2026 Courses:**\n  - [ECE 150 - Making Better Decisions by Humans and AI](https:\/\/www.probabilitycourse.com\/courses.php#ECE150) (4 credits)\n  - [ECE 214 - Probability and Statistics](https:\/\/www.probabilitycourse.com\/courses.php#ECE214) (4 credits at UMass Amherst)\n  - [ECE 579 - Math Tools for Data Science & Machine Learning](https:\/\/www.probabilitycourse.com\/courses.php#ECE579) (3 credits)\n  - [ECE 603 - Probability & Random Process](https:\/\/www.probabilitycourse.com\/courses.php#ECE603) (3 credits)\n- **A chapter on [simulation using Python](https:\/\/www.probabilitycourse.com\/chapter14\/chapter14.php) was added\\!**\n***\n## Welcome\nThis site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. This probability textbook can be used by both students and practitioners in engineering, mathematics, finance, and other related fields.\n\nThe site includes:\n\n- The entire textbook\n- Short video lectures that aid in learning the material\n- Online probability calculators for important functions and distributions\n- A solutions manual for instructors\n- [Lecture slides](https:\/\/www.dropbox.com\/sh\/iuhdrbq1yqf8t8q\/AAAV6npl5xFkqfvTd2jSEjjwa?dl=0)\n\nThe print version of the book is available through Amazon [here](https:\/\/www.amazon.com\/Introduction-Probability-Statistics-Random-Processes\/dp\/0990637204\/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik).\n\n[![Book Cover](https:\/\/www.probabilitycourse.com\/images\/home\/book_cover.jpg)](https:\/\/www.amazon.com\/Introduction-Probability-Statistics-Random-Processes\/dp\/0990637204\/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik)\n\n## Book Coverage\nThis probability and statistics textbook covers:\n\n- Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods\n- Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities\n- Limit theorems and convergence\n- Introduction to mathematical statistics, in particular, Bayesian and classical statistics\n- Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion\n- Simulation using MATLAB, R, and Python\n## How to cite\nYou can cite this textbook as:\n\nH. Pishro-Nik, \"Introduction to probability, statistics, and random processes\", available at [https:\/\/www.probabilitycourse.com](https:\/\/www.probabilitycourse.com\/), Kappa Research LLC, 2014.\n\n## Student’s Solutions Guide\nSince the textbook's initial publication, many requested the distribution of solutions to the problems in the textbook. We published the student’s solutions guide which includes guided solutions to the **odd-numbered** end-of-chapter problems.\n\nThis guide is available on Amazon in both [print](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes\/dp\/0990637212\/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) and [kindle](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes-ebook\/dp\/B01HBRRHLY\/ref=sr_1_3?ie=UTF8&qid=1466517931&sr=8-3&keywords=pishro-nik) format:\n\n[![Solution Book Cover](https:\/\/www.probabilitycourse.com\/images\/home\/solution_cover.png)](https:\/\/www.amazon.com\/Solutions-Introduction-Probability-Statistics-Processes\/dp\/0990637212\/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik)\n\n## About the Author\n[H. Pishro-Nik](https:\/\/websites.umass.edu\/pishro\/) is a professor in the Department of Electrical and Computer Engineering at the University of Massachusetts Amherst. He received his B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from Georgia Institute of Technology, all in Electrical and Computer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making.","meta_canonical":null}

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Boilerpipe Text	News: Online Spring 2026 Courses: ECE 150 - Making Better Decisions by Humans and AI (4 credits) ECE 214 - Probability and Statistics (4 credits at UMass Amherst) ECE 579 - Math Tools for Data Science & Machine Learning (3 credits) ECE 603 - Probability & Random Process (3 credits) New book release: Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI is now available on Amazon in print , Kindle , and audiobook formats. Interested readers can visit our website to download a free sample. Step-by-Step Solutions to Selected Problems in Signals & Systems by Hamid Saeedi, Hossein Pishro-Nik This book is available on Amazon . Interested students can view sample problems and sample solutions . A chapter on simulation using Python was added! Welcome This site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. This probability textbook can be used by both students and practitioners in engineering, mathematics, finance, and other related fields. The site includes: The entire textbook Short video lectures that aid in learning the material Online probability calculators for important functions and distributions A solutions manual for instructors Lecture slides The print version of the book is available through Amazon here . Book Coverage This probability and statistics textbook covers: Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities Limit theorems and convergence Introduction to mathematical statistics, in particular, Bayesian and classical statistics Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion Simulation using MATLAB, R, and Python How to cite You can cite this textbook as: H. Pishro-Nik, "Introduction to probability, statistics, and random processes", available at https://www.probabilitycourse.com , Kappa Research LLC, 2014. Student’s Solutions Guide Since the textbook's initial publication, many requested the distribution of solutions to the problems in the textbook. We published the student’s solutions guide which includes guided solutions to the odd-numbered end-of-chapter problems. This guide is available on Amazon in both print and kindle format: About the Author H. Pishro-Nik is a professor in the Department of Electrical and Computer Engineering at the University of Massachusetts Amherst. He received his B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from Georgia Institute of Technology, all in Electrical and Computer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making.
Markdown	- [HOME](https://www.probabilitycourse.com/) - [VIDEOS](https://www.probabilitycourse.com/videos/videos.php) - [CALCULATOR](https://www.probabilitycourse.com/calculator/calculator.php) - [COMMENTS](https://www.probabilitycourse.com/comments.php) - [COURSES](https://www.probabilitycourse.com/courses.php) - [FOR INSTRUCTOR](https://www.probabilitycourse.com/for_instructors.php) - [LOG IN](https://www.probabilitycourse.com/Login/mobile_login.php) [![](https://www.probabilitycourse.com/images/Template/logo_probability_website.png)](https://www.probabilitycourse.com/) Chapters Menu - [HOME](https://www.probabilitycourse.com/) - [VIDEOS](https://www.probabilitycourse.com/videos/videos.php) - [CALCULATOR](https://www.probabilitycourse.com/calculator/calculator.php) - [COMMENTS](https://www.probabilitycourse.com/comments.php) - [COURSES](https://www.probabilitycourse.com/courses.php) - [FOR INSTRUCTORS](https://www.probabilitycourse.com/for_instructors.php) - [Sign In]() [Forgot password?](https://www.probabilitycourse.com/Login/forgot_password.php) Open Menu - [0 Preface](https://www.probabilitycourse.com/preface.php) - [1 Basic Concepts]() - [1\.0 Introduction](https://www.probabilitycourse.com/chapter1/1_0_0_introduction.php) - [1\.1 Introduction]() - [1\.1.0 What Is Probability?](https://www.probabilitycourse.com/chapter1/1_1_0_what_is_probability.php) - [1\.1.1 Example](https://www.probabilitycourse.com/chapter1/1_1_1_example.php) - [1\.2 Review of Set Theory]() - [1\.2.0 Review](https://www.probabilitycourse.com/chapter1/1_2_0_review_set_theory.php) - [1\.2.1 Venn Diagrams](https://www.probabilitycourse.com/chapter1/1_2_1_venn.php) - [1\.2.2 Set Operations](https://www.probabilitycourse.com/chapter1/1_2_2_set_operations.php) - [1\.2.3 Cardinality](https://www.probabilitycourse.com/chapter1/1_2_3_cardinality.php) - [1\.2.4 Functions](https://www.probabilitycourse.com/chapter1/1_2_4_functions.php) - [1\.2.5 Solved Problems](https://www.probabilitycourse.com/chapter1/1_2_5_solved1.php) - [1\.3 Random Experiments and Probabilities]() - [1\.3.1 Random Experiments](https://www.probabilitycourse.com/chapter1/1_3_1_random_experiments.php) - [1\.3.2 Probability](https://www.probabilitycourse.com/chapter1/1_3_2_probability.php) - [1\.3.3 Finding Probabilities](https://www.probabilitycourse.com/chapter1/1_3_3_finding_probabilities.php) - [1\.3.4 Discrete Models](https://www.probabilitycourse.com/chapter1/1_3_4_discrete_models.php) - [1\.3.5 Continuous Models](https://www.probabilitycourse.com/chapter1/1_3_5_continuous_models.php) - [1\.3.6 Solved Problems](https://www.probabilitycourse.com/chapter1/1_3_6_solved2.php) - [1\.4 Conditional Probability]() - [1\.4.0 Conditional Probability](https://www.probabilitycourse.com/chapter1/1_4_0_conditional_probability.php) - [1\.4.1 Independence](https://www.probabilitycourse.com/chapter1/1_4_1_independence.php) - [1\.4.2 Law of Total Probability](https://www.probabilitycourse.com/chapter1/1_4_2_total_probability.php) - [1\.4.3 Bayes' Rule](https://www.probabilitycourse.com/chapter1/1_4_3_bayes_rule.php) - [1\.4.4 Conditional Independence](https://www.probabilitycourse.com/chapter1/1_4_4_conditional_independence.php) - [1\.4.5 Solved Problems](https://www.probabilitycourse.com/chapter1/1_4_5_solved3.php) - [1\.5 Problems]() - [1\.5.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter1/1_5_0_chapter1_problems.php) - [2 Combinatorics: Counting Methods]() - [2\.1 Combinatorics]() - [2\.1.0 Finding Probabilities with Counting Methods](https://www.probabilitycourse.com/chapter2/2_1_0_counting.php) - [2\.1.1 Ordered with Replacement](https://www.probabilitycourse.com/chapter2/2_1_1_ordered_with_replacement.php) - [2\.1.2 Ordered without Replacement](https://www.probabilitycourse.com/chapter2/2_1_2_ordered_without_replacement.php) - [2\.1.3 Unordered without Replacement](https://www.probabilitycourse.com/chapter2/2_1_3_unordered_without_replacement.php) - [2\.1.4 Unordered with Replacement](https://www.probabilitycourse.com/chapter2/2_1_4_unordered_with_replacement.php) - [2\.1.5 Solved Problems](https://www.probabilitycourse.com/chapter2/2_1_5_solved2_1.php) - [2\.2 Problems]() - [2\.2.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter2/2_3_0_chapter2_problems.php) - [3 Discrete Random Variables]() - [3\.1 Basic Concepts]() - [3\.1.1 Random Variables](https://www.probabilitycourse.com/chapter3/3_1_1_random_variables.php) - [3\.1.2 Discrete Random Variables](https://www.probabilitycourse.com/chapter3/3_1_2_discrete_random_var.php) - [3\.1.3 Probability Mass Function](https://www.probabilitycourse.com/chapter3/3_1_3_pmf.php) - [3\.1.4 Independent Random Variables](https://www.probabilitycourse.com/chapter3/3_1_4_independent_random_var.php) - [3\.1.5 Special Distributions](https://www.probabilitycourse.com/chapter3/3_1_5_special_discrete_distr.php) - [3\.1.6 Solved Problems](https://www.probabilitycourse.com/chapter3/3_1_6_solved3_1.php) - [3\.2 More about Discrete Random Variables]() - [3\.2.1 Cumulative Distribution Function](https://www.probabilitycourse.com/chapter3/3_2_1_cdf.php) - [3\.2.2 Expectation](https://www.probabilitycourse.com/chapter3/3_2_2_expectation.php) - [3\.2.3 Functions of Random Variables](https://www.probabilitycourse.com/chapter3/3_2_3_functions_random_var.php) - [3\.2.4 Variance](https://www.probabilitycourse.com/chapter3/3_2_4_variance.php) - [3\.2.5 Solved Problems](https://www.probabilitycourse.com/chapter3/3_2_5_solved3_2.php) - [3\.3 Problems]() - [3\.3.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter3/3_3_0_chapter3_problems.php) - [4 Continuous and Mixed Random Variables]() - [4\.0 Introduction](https://www.probabilitycourse.com/chapter4/4_0_0_intro.php) - [4\.1 Continuous Random Variables]() - [4\.1.0 Continuous Random Variables and their Distributions](https://www.probabilitycourse.com/chapter4/4_1_0_continuous_random_vars_distributions.php) - [4\.1.1 Probability Density Function](https://www.probabilitycourse.com/chapter4/4_1_1_pdf.php) - [4\.1.2 Expected Value and Variance](https://www.probabilitycourse.com/chapter4/4_1_2_expected_val_variance.php) - [4\.1.3 Functions of Continuous Random Variables](https://www.probabilitycourse.com/chapter4/4_1_3_functions_continuous_var.php) - [4\.1.4 Solved Problems](https://www.probabilitycourse.com/chapter4/4_1_4_solved4_1.php) - [4\.2 Special Distributions]() - [4\.2.1 Uniform Distribution](https://www.probabilitycourse.com/chapter4/4_2_1_uniform.php) - [4\.2.2 Exponential Distribution](https://www.probabilitycourse.com/chapter4/4_2_2_exponential.php) - [4\.2.3 Normal (Gaussian) Distribution](https://www.probabilitycourse.com/chapter4/4_2_3_normal.php) - [4\.2.4 Gamma Distribution](https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php) - [4\.2.5 Other Distributions](https://www.probabilitycourse.com/chapter4/4_2_5_other_distr.php) - [4\.2.6 Solved Problems](https://www.probabilitycourse.com/chapter4/4_2_6_solved4_2.php) - [4\.3 Mixed Random Variables]() - [4\.3.1 Mixed Random Variables](https://www.probabilitycourse.com/chapter4/4_3_1_mixed.php) - [4\.3.2 Using the Delta Function](https://www.probabilitycourse.com/chapter4/4_3_2_delta_function.php) - [4\.3.3 Solved Problems](https://www.probabilitycourse.com/chapter4/4_3_3_solved4_3.php) - [4\.4 Problems]() - [4\.4.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter4/4_4_0_chapter4_problems.php) - [5 Joint Distributions]() - [5\.1 Two Discrete Random Variables]() - [5\.1.0 Two Random Variables](https://www.probabilitycourse.com/chapter5/5_1_0_joint_distributions.php) - [5\.1.1 Joint Probability Mass Function (PMF)](https://www.probabilitycourse.com/chapter5/5_1_1_joint_pmf.php) - [5\.1.2 Joint Cumulative Distribution Function (CDF)](https://www.probabilitycourse.com/chapter5/5_1_2_joint_cdf.php) - [5\.1.3 Conditioning and Independence](https://www.probabilitycourse.com/chapter5/5_1_3_conditioning_independence.php) - [5\.1.4 Functions of Two Random Variables](https://www.probabilitycourse.com/chapter5/5_1_4_functions_two_variables.php) - [5\.1.5 Conditional Expectation](https://www.probabilitycourse.com/chapter5/5_1_5_conditional_expectation.php) - [5\.1.6 Solved Problems](https://www.probabilitycourse.com/chapter5/5_1_6_solved_prob.php) - [5\.2 Two Continuous Random Variables]() - [5\.2.0 Two Continuous Random Variables](https://www.probabilitycourse.com/chapter5/5_2_0_continuous_vars.php) - [5\.2.1 Joint Probability Density Function](https://www.probabilitycourse.com/chapter5/5_2_1_joint_pdf.php) - [5\.2.2 Joint Cumulative Distribution Function](https://www.probabilitycourse.com/chapter5/5_2_2_joint_cdf.php) - [5\.2.3 Conditioning and Independence](https://www.probabilitycourse.com/chapter5/5_2_3_conditioning_independence.php) - [5\.2.4 Functions of Two Continuous Random Variables](https://www.probabilitycourse.com/chapter5/5_2_4_functions.php) - [5\.2.5 Solved Problems](https://www.probabilitycourse.com/chapter5/5_2_5_solved_prob.php) - [5\.3 More Topics]() - [5\.3.1 Covariance and Correlation](https://www.probabilitycourse.com/chapter5/5_3_1_covariance_correlation.php) - [5\.3.2 Bivariate Normal Distribution](https://www.probabilitycourse.com/chapter5/5_3_2_bivariate_normal_dist.php) - [5\.3.3 Solved Problems](https://www.probabilitycourse.com/chapter5/5_3_3_solved_probs.php) - [5\.4 Problems]() - [5\.4.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter5/5_4_0_chapter_problems.php) - [6 Multiple Random Variables]() - [6\.0 Introduction](https://www.probabilitycourse.com/chapter6/6_0_0_intro.php) - [6\.1 Methods for More Than Two Random Variables]() - [6\.1.1 Joint Distributions and Independence](https://www.probabilitycourse.com/chapter6/6_1_1_joint_distributions_independence.php) - [6\.1.2 Sums of Random Variables](https://www.probabilitycourse.com/chapter6/6_1_2_sums_random_variables.php) - [6\.1.3 Moment Generating Functions](https://www.probabilitycourse.com/chapter6/6_1_3_moment_functions.php) - [6\.1.4 Characteristic Functions](https://www.probabilitycourse.com/chapter6/6_1_4_characteristic_functions.php) - [6\.1.5 Random Vectors](https://www.probabilitycourse.com/chapter6/6_1_5_random_vectors.php) - [6\.1.6 Solved Problems](https://www.probabilitycourse.com/chapter6/6_1_6_solved_probs.php) - [6\.2 Probability Bounds]() - [6\.2.0 Probability Bounds](https://www.probabilitycourse.com/chapter6/6_2_0_probability_bounds.php) - [6\.2.1 Union Bound and Extension](https://www.probabilitycourse.com/chapter6/6_2_1_union_bound_and_exten.php) - [6\.2.2 Markov Chebyshev Inequalities](https://www.probabilitycourse.com/chapter6/6_2_2_markov_chebyshev_inequalities.php) - [6\.2.3 Chernoff Bounds](https://www.probabilitycourse.com/chapter6/6_2_3_chernoff_bounds.php) - [6\.2.4 Cauchy Schwarz Inequality](https://www.probabilitycourse.com/chapter6/6_2_4_cauchy_schwarz.php) - [6\.2.5 Jensen's Inequality](https://www.probabilitycourse.com/chapter6/6_2_5_jensen's_inequality.php) - [6\.2.6 Solved Problems](https://www.probabilitycourse.com/chapter6/6_2_6_solved6_2.php) - [6\.3 Problems]() - [6\.3.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter6/6_3_0_chapter_problems.php) - [7 Limit Theorems and Convergence of Random Variables]() - [7\.0 Introduction](https://www.probabilitycourse.com/chapter7/7_0_0_intro.php) - [7\.1 Limit Theorems]() - [7\.1.0 Limit Theorems](https://www.probabilitycourse.com/chapter7/7_1_0_limit_theorems.php) - [7\.1.1 Law of Large Numbers](https://www.probabilitycourse.com/chapter7/7_1_1_law_of_large_numbers.php) - [7\.1.2 Central Limit Theorem (CLT)](https://www.probabilitycourse.com/chapter7/7_1_2_central_limit_theorem.php) - [7\.1.3 Solved Problems](https://www.probabilitycourse.com/chapter7/7_1_3_solved_probs.php) - [7\.2 Convergence of Random Variables]() - [7\.2.0 Convergence of Random Variables](https://www.probabilitycourse.com/chapter7/7_2_0_convergence_of_random_variables.php) - [7\.2.1 Convergence of Sequence of Numbers](https://www.probabilitycourse.com/chapter7/7_2_1_convergence_of_a_seq_of_nums.php) - [7\.2.2 Sequence of Random Variables](https://www.probabilitycourse.com/chapter7/7_2_2_sequence_of_random_variables.php) - [7\.2.3 Different Types of Convergence for Sequences of Random Variables](https://www.probabilitycourse.com/chapter7/7_2_3_different_types_of_convergence_for_sequences_of_random_variables.php) - [7\.2.4 Convergence in Distribution](https://www.probabilitycourse.com/chapter7/7_2_4_convergence_in_distribution.php) - [7\.2.5 Convergence in Probability](https://www.probabilitycourse.com/chapter7/7_2_5_convergence_in_probability.php) - [7\.2.6 Convergence in Mean](https://www.probabilitycourse.com/chapter7/7_2_6_convergence_in_mean.php) - [7\.2.7 Almost Sure Convergence](https://www.probabilitycourse.com/chapter7/7_2_7_almost_sure_convergence.php) - [7\.2.8 Solved Problems](https://www.probabilitycourse.com/chapter7/7_2_8_solved_probs.php) - [7\.3 Problems]() - [7\.3.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter7/7_3_0_chapter_problem.php) - [8 Statistical Inference I: Classical Methods]() - [8\.1 Introduction]() - [8\.1.0 Introduction](https://www.probabilitycourse.com/chapter8/8_1_0_intro.php) - [8\.1.1 Random Sampling](https://www.probabilitycourse.com/chapter8/8_1_1_random_sampling.php) - [8\.2 Point Estimation]() - [8\.2.0 Point Estimation](https://www.probabilitycourse.com/chapter8/8_2_0_point_estimation.php) - [8\.2.1 Evaluating Estimators](https://www.probabilitycourse.com/chapter8/8_2_1_evaluating_estimators.php) - [8\.2.2 Point Estimators for Mean and Variance](https://www.probabilitycourse.com/chapter8/8_2_2_point_estimators_for_mean_and_var.php) - [8\.2.3 Maximum Likelihood Estimation (MLE)](https://www.probabilitycourse.com/chapter8/8_2_3_max_likelihood_estimation.php) - [8\.2.4 Asymptotic Properties of MLEs](https://www.probabilitycourse.com/chapter8/8_2_4_asymptotic_probs_of_MLE.php) - [8\.2.5 Solved Problems](https://www.probabilitycourse.com/chapter8/8_2_5_solved_probs.php) - [8\.3 Interval Estimation (Confidence Intervals)]() - [8\.3.0 Interval Estimation (Confidence Intervals)](https://www.probabilitycourse.com/chapter8/8_3_0_interval_estimation.php) - [8\.3.1 The general framework of Interval Estimation](https://www.probabilitycourse.com/chapter8/8_3_1_gen_framework_of_int_estimation.php) - [8\.3.2 Finding Interval Estimators](https://www.probabilitycourse.com/chapter8/8_3_2_finding_interval_estimators.php) - [8\.3.3 Confidence Intervals for Normal Samples](https://www.probabilitycourse.com/chapter8/8_3_3_confidence_intervals_for_norm_samples.php) - [8\.3.4 Solved Problems](https://www.probabilitycourse.com/chapter8/8_3_4_solved_probs.php) - [8\.4 Hypothesis Testing]() - [8\.4.1 Introduction](https://www.probabilitycourse.com/chapter8/8_4_1_intro.php) - [8\.4.2 General Setting and Definitions](https://www.probabilitycourse.com/chapter8/8_4_2_general_setting_definitions.php) - [8\.4.3 Hypothesis Testing for the Mean](https://www.probabilitycourse.com/chapter8/8_4_3_hypothesis_testing_for_mean.php) - [8\.4.4 P-Values](https://www.probabilitycourse.com/chapter8/8_4_4_p_vals.php) - [8\.4.5 Likelihood Ratio Tests](https://www.probabilitycourse.com/chapter8/8_4_5_likelihood_ratio_tests.php) - [8\.4.6 Solved Problems](https://www.probabilitycourse.com/chapter8/8_4_6_solved_probs.php) - [8\.5 Linear Regression]() - [8\.5.0 Linear Regression](https://www.probabilitycourse.com/chapter8/8_5_0_linear_regression.php) - [8\.5.1 Simple Linear Regression Model](https://www.probabilitycourse.com/chapter8/8_5_1_simple_linear_regression_model.php) - [8\.5.2 The First Method for Finding beta](https://www.probabilitycourse.com/chapter8/8_5_2_first_method_for_finding_beta.php) - [8\.5.3 The Method of Least Squares](https://www.probabilitycourse.com/chapter8/8_5_3_the_method_of_least_squares.php) - [8\.5.4 Extensions and Issues](https://www.probabilitycourse.com/chapter8/8_5_4_extensions_and_issues.php) - [8\.5.5 Solved Problems](https://www.probabilitycourse.com/chapter8/8_5_5_solved_probs.php) - [8\.6 Problems]() - [8\.6.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter8/8_6_0_ch_probs.php) - [9 Statistical Inference II: Bayesian Inference]() - [9\.1 Bayesian Inference]() - [9\.1.0 Bayesian Inference](https://www.probabilitycourse.com/chapter9/9_1_0_bayesian_inference.php) - [9\.1.1 Prior and Posterior](https://www.probabilitycourse.com/chapter9/9_1_1_prior_and_posterior.php) - [9\.1.2 Maximum A Posteriori (MAP) Estimation](https://www.probabilitycourse.com/chapter9/9_1_2_MAP_estimation.php) - [9\.1.3 Comparison to ML Estimation](https://www.probabilitycourse.com/chapter9/9_1_3_comparison_to_ML_estimation.php) - [9\.1.4 Conditional Expectation (MMSE)](https://www.probabilitycourse.com/chapter9/9_1_4_conditional_expectation_MMSE.php) - [9\.1.5 Mean Squared Error (MSE)](https://www.probabilitycourse.com/chapter9/9_1_5_mean_squared_error_MSE.php) - [9\.1.6 Linear MMSE Estimation of Random Variables](https://www.probabilitycourse.com/chapter9/9_1_6_linear_MMSE_estimat_of_random_vars.php) - [9\.1.7 Estimation for Random Vectors](https://www.probabilitycourse.com/chapter9/9_1_7_estimation_for_random_vectors.php) - [9\.1.8 Bayesian Hypothesis Testing](https://www.probabilitycourse.com/chapter9/9_1_8_bayesian_hypothesis_testing.php) - [9\.1.9 Bayesian Interval Estimation](https://www.probabilitycourse.com/chapter9/9_1_9_bayesian_interval_estimation.php) - [9\.1.10 Solved Problems](https://www.probabilitycourse.com/chapter9/9_1_10_solved_probs.php) - [9\.2 Problems]() - [9\.2.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter9/9_2_0_ch_probs.php) - [10 Introduction to Random Processes]() - [10\.1 Basic Concepts]() - [10\.1.0 Basic Concepts](https://www.probabilitycourse.com/chapter10/10_1_0_basic_concepts.php) - [10\.1.1 PDFs and CDFs](https://www.probabilitycourse.com/chapter10/10_1_1_PDFs_and_CDFs.php) - [10\.1.2 Mean and Correlation Functions](https://www.probabilitycourse.com/chapter10/10_1_2_mean_and_correlation_functions.php) - [10\.1.3 Multiple Random Processes](https://www.probabilitycourse.com/chapter10/10_1_3_multiple_random_processes.php) - [10\.1.4 Stationary Processes](https://www.probabilitycourse.com/chapter10/10_1_4_stationary_processes.php) - [10\.1.5 Gaussian Random Processes](https://www.probabilitycourse.com/chapter10/10_1_5_gaussian_random_processes.php) - [10\.1.6 Solved Problems](https://www.probabilitycourse.com/chapter10/10_1_6_solved_probs.php) - [10\.2 Processing of Random Signals]() - [10\.2.0 Processing of Random Signals](https://www.probabilitycourse.com/chapter10/10_2_0_processing_of_random_signals.php) - [10\.2.1 Power Spectral Density](https://www.probabilitycourse.com/chapter10/10_2_1_power_spectral_density.php) - [10\.2.2 Linear Time-Invariant (LTI) Systems with Random Inputs](https://www.probabilitycourse.com/chapter10/10_2_2_LTI_systems_with_random_inputs.php) - [10\.2.3 Power in a Frequency Band](https://www.probabilitycourse.com/chapter10/10_2_3_power_in_a_frequency_band.php) - [10\.2.4 White Noise](https://www.probabilitycourse.com/chapter10/10_2_4_white_noise.php) - [10\.2.5 Solved Problems](https://www.probabilitycourse.com/chapter10/10_2_5_solved_probs.php) - [10\.3 Problems]() - [10\.3.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter10/10_3_0_ch_probs.php) - [11 Some Important Random Processes]() - [11\.1 Poisson Processes]() - [11\.1.0 Introduction](https://www.probabilitycourse.com/chapter11/11_0_0_intro.php) - [11\.1.1 Counting Processes](https://www.probabilitycourse.com/chapter11/11_1_1_counting_processes.php) - [11\.1.2 Basic Concepts of the Poisson Process](https://www.probabilitycourse.com/chapter11/11_1_2_basic_concepts_of_the_poisson_process.php) - [11\.1.3 Merging and Splitting Poisson Processes](https://www.probabilitycourse.com/chapter11/11_1_3_merging_and_splitting_poisson_processes.php) - [11\.1.4 Nonhomogeneous Poisson Processes](https://www.probabilitycourse.com/chapter11/11_1_4_nonhomogeneous_poisson_processes.php) - [11\.1.5 Solved Problems](https://www.probabilitycourse.com/chapter11/11_1_5_solved_probs.php) - [11\.2 Discrete-Time Markov Chains]() - [11\.2.1 Introduction](https://www.probabilitycourse.com/chapter11/11_2_1_introduction.php) - [11\.2.2 State Transition Matrix and Diagram](https://www.probabilitycourse.com/chapter11/11_2_2_state_transition_matrix_and_diagram.php) - [11\.2.3 Probability Distributions](https://www.probabilitycourse.com/chapter11/11_2_3_probability_distributions.php) - [11\.2.4 Classification of States](https://www.probabilitycourse.com/chapter11/11_2_4_classification_of_states.php) - [11\.2.5 Using the Law of Total Probability with Recursion](https://www.probabilitycourse.com/chapter11/11_2_5_using_the_law_of_total_probability_with_recursion.php) - [11\.2.6 Stationary and Limiting Distributions](https://www.probabilitycourse.com/chapter11/11_2_6_stationary_and_limiting_distributions.php) - [11\.2.7 Solved Problems](https://www.probabilitycourse.com/chapter11/11_2_7_solved_probs.php) - [11\.3 Continuous-Time Markov Chains]() - [11\.3.1 Introduction](https://www.probabilitycourse.com/chapter11/11_3_1_introduction.php) - [11\.3.2 Stationary and Limiting Distributions](https://www.probabilitycourse.com/chapter11/11_3_2_stationary_and_limiting_distributions.php) - [11\.3.3 The Generator Matrix](https://www.probabilitycourse.com/chapter11/11_3_3_the_generator_matrix.php) - [11\.3.4 Solved Problems](https://www.probabilitycourse.com/chapter11/11_3_4_solved_probs.php) - [11\.4 Brownian Motion (Wiener Process)]() - [11\.4.0 Brownian Motion (Wiener Process)](https://www.probabilitycourse.com/chapter11/11_4_0_brownian_motion_wiener_process.php) - [11\.4.1 Brownian Motion as the Limit of a Symmetric Random Walk](https://www.probabilitycourse.com/chapter11/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php) - [1\.4.2 Definition and Some Properties](https://www.probabilitycourse.com/chapter11/11_4_2_definition_and_some_properties.php) - [11\.4.3 Solved Problems](https://www.probabilitycourse.com/chapter11/11_4_3_solved_probs.php) - [11\.5 Problems]() - [11\.5.0 End of Chapter Problems](https://www.probabilitycourse.com/chapter11/11_5_0_end_of_chapter_problems.php) - [12 Introduction to Simulation Using MATLAB](https://www.probabilitycourse.com/chapter12/chapter12.php) - [13 Introduction to Simulation Using R](https://www.probabilitycourse.com/chapter13/chapter13.php) - [14 Introduction to Simulation Using Python](https://www.probabilitycourse.com/chapter14/chapter14.php) - [15 Recursive Methods](https://www.probabilitycourse.com/chapter15/chapter15.php) - [Appendix]() - [Some Important Distributions](https://www.probabilitycourse.com/appendix/some_important_distributions.php) - [Review of the Fourier Transform](https://www.probabilitycourse.com/appendix/review_fourier_transform.php) - [Bibliography](https://www.probabilitycourse.com/bibliography.php) ## News: - Online Spring 2026 Courses: - [ECE 150 - Making Better Decisions by Humans and AI](https://www.probabilitycourse.com/courses.php#ECE150) (4 credits) - [ECE 214 - Probability and Statistics](https://www.probabilitycourse.com/courses.php#ECE214) (4 credits at UMass Amherst) - [ECE 579 - Math Tools for Data Science & Machine Learning](https://www.probabilitycourse.com/courses.php#ECE579) (3 credits) - [ECE 603 - Probability & Random Process](https://www.probabilitycourse.com/courses.php#ECE603) (3 credits) - A chapter on [simulation using Python](https://www.probabilitycourse.com/chapter14/chapter14.php) was added\! * # Welcome This site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. This probability textbook can be used by both students and practitioners in engineering, mathematics, finance, and other related fields. The site includes: - The entire textbook - Short video lectures that aid in learning the material - Online probability calculators for important functions and distributions - A solutions manual for instructors - [Lecture slides](https://www.dropbox.com/sh/iuhdrbq1yqf8t8q/AAAV6npl5xFkqfvTd2jSEjjwa?dl=0) The print version of the book is available through Amazon [here](https://www.amazon.com/Introduction-Probability-Statistics-Random-Processes/dp/0990637204/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik). [![Book Cover](https://www.probabilitycourse.com/images/home/book_cover.jpg)](https://www.amazon.com/Introduction-Probability-Statistics-Random-Processes/dp/0990637204/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik) # Book Coverage This probability and statistics textbook covers: - Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods - Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities - Limit theorems and convergence - Introduction to mathematical statistics, in particular, Bayesian and classical statistics - Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion - Simulation using MATLAB, R, and Python # How to cite You can cite this textbook as: H. Pishro-Nik, "Introduction to probability, statistics, and random processes", available at <https://www.probabilitycourse.com>, Kappa Research LLC, 2014. # Student’s Solutions Guide Since the textbook's initial publication, many requested the distribution of solutions to the problems in the textbook. We published the student’s solutions guide which includes guided solutions to the odd-numbered** end-of-chapter problems. This guide is available on Amazon in both [print](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes/dp/0990637212/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) and [kindle](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes-ebook/dp/B01HBRRHLY/ref=sr_1_3?ie=UTF8&qid=1466517931&sr=8-3&keywords=pishro-nik) format: [![Solution Book Cover](https://www.probabilitycourse.com/images/home/solution_cover.png)](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes/dp/0990637212/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) # About the Author [H. Pishro-Nik](https://websites.umass.edu/pishro/) is a professor in the Department of Electrical and Computer Engineering at the University of Massachusetts Amherst. He received his B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from Georgia Institute of Technology, all in Electrical and Computer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making. [![Creative Commons License](https://i.creativecommons.org/l/by-nc-nd/3.0/88x31.png)](https://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US) Introduction to Probability by [Hossein Pishro-Nik](https://websites.umass.edu/pishro/) is licensed under a [Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License](https://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US)
Readable Markdown	## News: - Online Spring 2026 Courses: - [ECE 150 - Making Better Decisions by Humans and AI](https://www.probabilitycourse.com/courses.php#ECE150) (4 credits) - [ECE 214 - Probability and Statistics](https://www.probabilitycourse.com/courses.php#ECE214) (4 credits at UMass Amherst) - [ECE 579 - Math Tools for Data Science & Machine Learning](https://www.probabilitycourse.com/courses.php#ECE579) (3 credits) - [ECE 603 - Probability & Random Process](https://www.probabilitycourse.com/courses.php#ECE603) (3 credits) - A chapter on [simulation using Python](https://www.probabilitycourse.com/chapter14/chapter14.php) was added\! * ## Welcome This site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. This probability textbook can be used by both students and practitioners in engineering, mathematics, finance, and other related fields. The site includes: - The entire textbook - Short video lectures that aid in learning the material - Online probability calculators for important functions and distributions - A solutions manual for instructors - [Lecture slides](https://www.dropbox.com/sh/iuhdrbq1yqf8t8q/AAAV6npl5xFkqfvTd2jSEjjwa?dl=0) The print version of the book is available through Amazon [here](https://www.amazon.com/Introduction-Probability-Statistics-Random-Processes/dp/0990637204/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik). [![Book Cover](https://www.probabilitycourse.com/images/home/book_cover.jpg)](https://www.amazon.com/Introduction-Probability-Statistics-Random-Processes/dp/0990637204/ref=sr_1_1?ie=UTF8&qid=1408880878&sr=8-1&keywords=pishro-nik) ## Book Coverage This probability and statistics textbook covers: - Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods - Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities - Limit theorems and convergence - Introduction to mathematical statistics, in particular, Bayesian and classical statistics - Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion - Simulation using MATLAB, R, and Python ## How to cite You can cite this textbook as: H. Pishro-Nik, "Introduction to probability, statistics, and random processes", available at [https://www.probabilitycourse.com](https://www.probabilitycourse.com/), Kappa Research LLC, 2014. ## Student’s Solutions Guide Since the textbook's initial publication, many requested the distribution of solutions to the problems in the textbook. We published the student’s solutions guide which includes guided solutions to the odd-numbered** end-of-chapter problems. This guide is available on Amazon in both [print](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes/dp/0990637212/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) and [kindle](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes-ebook/dp/B01HBRRHLY/ref=sr_1_3?ie=UTF8&qid=1466517931&sr=8-3&keywords=pishro-nik) format: [![Solution Book Cover](https://www.probabilitycourse.com/images/home/solution_cover.png)](https://www.amazon.com/Solutions-Introduction-Probability-Statistics-Processes/dp/0990637212/ref=sr_1_2?ie=UTF8&qid=1466517931&sr=8-2&keywords=pishro-nik) ## About the Author [H. Pishro-Nik](https://websites.umass.edu/pishro/) is a professor in the Department of Electrical and Computer Engineering at the University of Massachusetts Amherst. He received his B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from Georgia Institute of Technology, all in Electrical and Computer Engineering. His research interests include Information Theory, Privacy, Autonomous Agents, Generative AI, and Decision Making.
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