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1. Shard Calculation

Query:

Response:

Calculated Shard: 115 (from laksa132)

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/chapter11/11_4_0_brownian_motion_wiener_process.php\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.probabilitycourse.com/chapter11/11_4_0_brownian_motion_wiener_process.php\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.probabilitycourse.com\/chapter11\/11_4_0_brownian_motion_wiener_process.php","crawl_time":1776720741,"first_indexed_time":1513404301,"http_code":200,"src_unparsed":"com,probabilitycourse!www,\/chapter11\/11_4_0_brownian_motion_wiener_process.php s443","src_root_hash":"8531609719846996715","history_drop_reason":null,"meta_title":"Brownian Motion (Wiener Process)","meta_descriptions":[""],"attrs_boilerpipe_text":"Brownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion.\nFigure 11.29 - A possible realization of Brownian motion.\nIn this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand\nIt\no\n¯\ncalculus\n, a topic that is beyond the scope of this book. A good place to start learning\nI\nt\no\n¯\ncalculus is\n[25]\n.\nThe print version of the book is available on\nAmazon\n.\nPractical uncertainty:\nUseful Ideas in Decision-Making, Risk, Randomness, & AI","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\n[←]() [previous](https:\/\/www.probabilitycourse.com\/chapter11\/chapter11\/11_3_4_solved_probs.php)\n\n[next](https:\/\/www.probabilitycourse.com\/chapter11\/chapter11\/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php) [→]()\n***\n## 11\\.4.0 Brownian Motion (Wiener Process)\nBrownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion.\n\n![BM](https:\/\/www.probabilitycourse.com\/chapter11\/images\/chapter11\/BM_b.png)\n\nFigure 11.29 - A possible realization of Brownian motion.\n\nIn this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand\n\nIto¯¯¯\n\nIt\n\no\n\n¯\n\n*calculus*, a topic that is beyond the scope of this book. A good place to start learning\n\nIto¯¯¯\n\nI\n\nt\n\no\n\n¯\n\ncalculus is [\\[25\\]](https:\/\/www.probabilitycourse.com\/bibliography.php#BM-calculus08).\n\n[←]() [previous](https:\/\/www.probabilitycourse.com\/chapter11\/chapter11\/11_3_4_solved_probs.php)\n\n[next](https:\/\/www.probabilitycourse.com\/chapter11\/chapter11\/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php) [→]()\n\n***\n|   |\n|---|\n| The print version of the book is available on [Amazon](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) |\n| **Practical uncertainty:** *Useful Ideas in Decision-Making, Risk, Randomness, & AI* [![ractical Uncertaintly Cover](https:\/\/www.probabilitycourse.com\/images\/home\/practical_uncertaintly.PNG)](https:\/\/www.amazon.com\/dp\/B0CH2BHRVH\/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1693837152&sr=8-1) |\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[![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":"***\nBrownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion.\n\n![BM](https:\/\/www.probabilitycourse.com\/images\/chapter11\/BM_b.png)\n\nFigure 11.29 - A possible realization of Brownian motion.\n\nIn this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand It o ¯ *calculus*, a topic that is beyond the scope of this book. A good place to start learning I t o ¯ calculus is [\\[25\\]](https:\/\/www.probabilitycourse.com\/bibliography.php#BM-calculus08).\n\n***\n|   |\n|---|\n| The print version of the book is available on [Amazon](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) |\n| **Practical uncertainty:** *Useful Ideas in Decision-Making, Risk, Randomness, & AI* [![ractical Uncertaintly Cover](https:\/\/www.probabilitycourse.com\/images\/home\/practical_uncertaintly.PNG)](https:\/\/www.amazon.com\/dp\/B0CH2BHRVH\/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1693837152&sr=8-1) |","meta_canonical":null}

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Boilerpipe Text	Brownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion. Figure 11.29 - A possible realization of Brownian motion. In this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand It o ¯ calculus , a topic that is beyond the scope of this book. A good place to start learning I t o ¯ calculus is [25] . The print version of the book is available on Amazon . Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI
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) [←]() [previous](https://www.probabilitycourse.com/chapter11/chapter11/11_3_4_solved_probs.php) [next](https://www.probabilitycourse.com/chapter11/chapter11/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php) [→]() *** ## 11\.4.0 Brownian Motion (Wiener Process) Brownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion. ![BM](https://www.probabilitycourse.com/chapter11/images/chapter11/BM_b.png) Figure 11.29 - A possible realization of Brownian motion. In this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand Ito¯¯¯ It o ¯ calculus, a topic that is beyond the scope of this book. A good place to start learning Ito¯¯¯ I t o ¯ calculus is [\[25\]](https://www.probabilitycourse.com/bibliography.php#BM-calculus08). [←]() [previous](https://www.probabilitycourse.com/chapter11/chapter11/11_3_4_solved_probs.php) [next](https://www.probabilitycourse.com/chapter11/chapter11/11_4_1_brownian_motion_as_the_limit_of_a_symmetric_random_walk.php) [→]() * \| \| \|---\| \| The print version of the book is available on [Amazon](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) \| \| Practical uncertainty:** Useful Ideas in Decision-Making, Risk, Randomness, & AI [![ractical Uncertaintly Cover](https://www.probabilitycourse.com/images/home/practical_uncertaintly.PNG)](https://www.amazon.com/dp/B0CH2BHRVH/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1693837152&sr=8-1) \| 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) [![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	*** Brownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion. ![BM](https://www.probabilitycourse.com/images/chapter11/BM_b.png) Figure 11.29 - A possible realization of Brownian motion. In this section, we provide a very brief introduction to Brownian motion. It is worth noting that in order to have a deep understanding of Brownian motion, one needs to understand It o ¯ calculus, a topic that is beyond the scope of this book. A good place to start learning I t o ¯ calculus is [\[25\]](https://www.probabilitycourse.com/bibliography.php#BM-calculus08). * \| \| \|---\| \| The print version of the book is available on [Amazon](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) \| \| Practical uncertainty:** Useful Ideas in Decision-Making, Risk, Randomness, & AI [![ractical Uncertaintly Cover](https://www.probabilitycourse.com/images/home/practical_uncertaintly.PNG)](https://www.amazon.com/dp/B0CH2BHRVH/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1693837152&sr=8-1) \|
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