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## Probability ### The Analysis of Data, volume 1 - [0](http://theanalysisofdata.com/probability/3_12.html) - Front Matter - [0\.1: Contents](http://theanalysisofdata.com/probability/0_1.html) - [0\.2: Preface](http://theanalysisofdata.com/probability/0_2.html) - [1](http://theanalysisofdata.com/probability/3_12.html) - Basic Definitions - [1\.1: Sample Space and Events](http://theanalysisofdata.com/probability/1_1.html) - [1\.2: The Probability Function](http://theanalysisofdata.com/probability/1_2.html) - [1\.3: Classical Model 1](http://theanalysisofdata.com/probability/1_3.html) - [1\.4: Classical Model 2](http://theanalysisofdata.com/probability/1_4.html) - [1\.5: Conditional Probability](http://theanalysisofdata.com/probability/1_5.html) - [1\.6: Basic Combinatorics](http://theanalysisofdata.com/probability/1_6.html) - [1\.7: Probability and Measure](http://theanalysisofdata.com/probability/1_7.html) - [1\.8: Notes](http://theanalysisofdata.com/probability/1_8.html) - [1\.9: Exercises](http://theanalysisofdata.com/probability/1_9.html) - [2](http://theanalysisofdata.com/probability/3_12.html) - Random Variables - [2\.1: Basic Definitions](http://theanalysisofdata.com/probability/2_1.html) - [2\.2: Functions of RVs](http://theanalysisofdata.com/probability/2_2.html) - [2\.3: Expectation and Variance](http://theanalysisofdata.com/probability/2_3.html) - [2\.4: Moments and MGF](http://theanalysisofdata.com/probability/2_4.html) - [2\.5: RVs and Measure Theory](http://theanalysisofdata.com/probability/2_5.html) - [2\.6: Notes](http://theanalysisofdata.com/probability/2_6.html) - [2\.7: Exercises](http://theanalysisofdata.com/probability/2_7.html) - [3](http://theanalysisofdata.com/probability/3_12.html) - Important RVs - [3\.1: Bernoulli RV](http://theanalysisofdata.com/probability/3_1.html) - [3\.2: Binomial RV](http://theanalysisofdata.com/probability/3_2.html) - [3\.3: Geometric RV](http://theanalysisofdata.com/probability/3_3.html) - [3\.4: Hypergeometric RV](http://theanalysisofdata.com/probability/3_4.html) - [3\.5: Negative Binomial RV](http://theanalysisofdata.com/probability/3_5.html) - [3\.6: Poisson RV](http://theanalysisofdata.com/probability/3_6.html) - [3\.7: Uniform RV](http://theanalysisofdata.com/probability/3_7.html) - [3\.8: Exponential RV](http://theanalysisofdata.com/probability/3_8.html) - [3\.9: Gaussian RV](http://theanalysisofdata.com/probability/3_9.html) - [3\.10: Gamma RV](http://theanalysisofdata.com/probability/3_10.html) - [3\.11: t RV](http://theanalysisofdata.com/probability/3_11.html) - [3\.12: Beta RV](http://theanalysisofdata.com/probability/3_12.html) - [3\.13: Mixture RV](http://theanalysisofdata.com/probability/3_13.html) - [3\.14: Empirical RV](http://theanalysisofdata.com/probability/3_14.html) - [3\.15: Smoothed Empirical RV](http://theanalysisofdata.com/probability/3_15.html) - [3\.16: Notes](http://theanalysisofdata.com/probability/3_16.html) - [3\.17: Exercises](http://theanalysisofdata.com/probability/3_17.html) - [4](http://theanalysisofdata.com/probability/3_12.html) - Random Vectors - [4\.1: Basic Definitions](http://theanalysisofdata.com/probability/4_1.html) - [4\.2: Joint Pmf, Pdf, and Cdf](http://theanalysisofdata.com/probability/4_2.html) - [4\.3: Marginal Random Vectors](http://theanalysisofdata.com/probability/4_3.html) - [4\.4: Functions of Random Vectors](http://theanalysisofdata.com/probability/4_4.html) - [4\.5: Conditional Random Vectors](http://theanalysisofdata.com/probability/4_5.html) - [4\.6: Moments](http://theanalysisofdata.com/probability/4_6.html) - [4\.7: Conditional Expectation](http://theanalysisofdata.com/probability/4_7.html) - [4\.8: Moment Generating Functions](http://theanalysisofdata.com/probability/4_8.html) - [4\.9: Random Vectors and Measure](http://theanalysisofdata.com/probability/4_9.html) - [4\.10: Notes](http://theanalysisofdata.com/probability/4_10.html) - [4\.11: Exercises](http://theanalysisofdata.com/probability/4_11.html) - [5](http://theanalysisofdata.com/probability/3_12.html) - Important Vectors - [5\.1: Multinomial Vectors](http://theanalysisofdata.com/probability/5_1.html) - [5\.2: Gaussian Vectors](http://theanalysisofdata.com/probability/5_2.html) - [5\.3: Dirichlet Vectors](http://theanalysisofdata.com/probability/5_3.html) - [5\.4: Mixture Vectors](http://theanalysisofdata.com/probability/5_4.html) - [5\.5: Exponential Family](http://theanalysisofdata.com/probability/5_5.html) - [5\.6: Notes](http://theanalysisofdata.com/probability/5_6.html) - [5\.7: Exercises](http://theanalysisofdata.com/probability/5_7.html) - [6](http://theanalysisofdata.com/probability/3_12.html) - Random Processes - [6\.1: Basic Definitions](http://theanalysisofdata.com/probability/6_1.html) - [6\.2: Marginals](http://theanalysisofdata.com/probability/6_2.html) - [6\.3: Moments](http://theanalysisofdata.com/probability/6_3.html) - [6\.4: Random Walk](http://theanalysisofdata.com/probability/6_4.html) - [6\.5: Processes and Measure](http://theanalysisofdata.com/probability/6_5.html) - [6\.6: Borell-Cantelli and Zero-One](http://theanalysisofdata.com/probability/6_6.html) - [6\.7: Notes](http://theanalysisofdata.com/probability/6_7.html) - [6\.8: Exercises](http://theanalysisofdata.com/probability/6_8.html) - [7](http://theanalysisofdata.com/probability/3_12.html) - Important RPs - [7\.1: Markov Chains](http://theanalysisofdata.com/probability/7_1.html) - [7\.2: Poisson Process](http://theanalysisofdata.com/probability/7_2.html) - [7\.3: Gaussian Process](http://theanalysisofdata.com/probability/7_3.html) - [7\.4: Notes](http://theanalysisofdata.com/probability/7_4.html) - [7\.5: Exercises](http://theanalysisofdata.com/probability/7_5.html) - [8](http://theanalysisofdata.com/probability/3_12.html) - Limit Theorems - [8\.1: Modes of Convergence](http://theanalysisofdata.com/probability/8_1.html) - [8\.2: Relationship of Modes](http://theanalysisofdata.com/probability/8_2.html) - [8\.3: DCT Theorem for Vectors](http://theanalysisofdata.com/probability/8_3.html) - [8\.4: Scheffe's Theorem](http://theanalysisofdata.com/probability/8_4.html) - [8\.5: Portmanteau Lemma](http://theanalysisofdata.com/probability/8_5.html) - [8\.6: Law of Large Numbers](http://theanalysisofdata.com/probability/8_6.html) - [8\.7: Characteristic Functions](http://theanalysisofdata.com/probability/8_7.html) - [8\.8: Levy's Theorem](http://theanalysisofdata.com/probability/8_8.html) - [8\.9: Central Limit Theorem](http://theanalysisofdata.com/probability/8_9.html) - [8\.10: Continuous Mapping Theorem](http://theanalysisofdata.com/probability/8_10.html) - [8\.11: Slustsky's Theorem](http://theanalysisofdata.com/probability/8_11.html) - [8\.12: Notes](http://theanalysisofdata.com/probability/8_12.html) - [8\.13: Exercises](http://theanalysisofdata.com/probability/8_13.html) - [A](http://theanalysisofdata.com/probability/3_12.html) - Set Theory - [A.1: Basic Definition](http://theanalysisofdata.com/probability/A_1.html) - [A.2: Functions](http://theanalysisofdata.com/probability/A_2.html) - [A.3: Cardinality](http://theanalysisofdata.com/probability/A_3.html) - [A.4: Limits of Sets](http://theanalysisofdata.com/probability/A_4.html) - [A.5: Notes](http://theanalysisofdata.com/probability/A_5.html) - [A.6: Exercises](http://theanalysisofdata.com/probability/A_6.html) - [B](http://theanalysisofdata.com/probability/3_12.html) - Metric Spaces - [B.1: Basic Definitions](http://theanalysisofdata.com/probability/B_1.html) - [B.2: Limits](http://theanalysisofdata.com/probability/B_2.html) - [B.3: Continuity](http://theanalysisofdata.com/probability/B_3.html) - [B.4: Euclidean Space](http://theanalysisofdata.com/probability/B_4.html) - [B.5: Growth of Functions](http://theanalysisofdata.com/probability/B_5.html) - [B.6: Notes](http://theanalysisofdata.com/probability/B_6.html) - [B.7: Exercises](http://theanalysisofdata.com/probability/B_7.html) - [C](http://theanalysisofdata.com/probability/3_12.html) - Linear Algebra - [C.1: Basic Definitions](http://theanalysisofdata.com/probability/C_1.html) - [C.2: Rank](http://theanalysisofdata.com/probability/C_2.html) - [C.3: Eigenvalues and Determinant](http://theanalysisofdata.com/probability/C_3.html) - [C.4: Semidefinite Matrices](http://theanalysisofdata.com/probability/C_4.html) - [C.5: SVD](http://theanalysisofdata.com/probability/C_5.html) - [C.6: Notes](http://theanalysisofdata.com/probability/C_6.html) - [C.7: Exercises](http://theanalysisofdata.com/probability/C_7.html) - [D](http://theanalysisofdata.com/probability/3_12.html) - Differentiation - [D.1: Scalar Differentiation](http://theanalysisofdata.com/probability/D_1.html) - [D.2: Power and Taylor Series](http://theanalysisofdata.com/probability/D_2.html) - [D.3: Notes](http://theanalysisofdata.com/probability/D_3.html) - [D.4: Exercises](http://theanalysisofdata.com/probability/D_4.html) - [E](http://theanalysisofdata.com/probability/3_12.html) - Measure Theory - [E.1: Sigma Algebras](http://theanalysisofdata.com/probability/E_1.html) - [E.2: Measure Function](http://theanalysisofdata.com/probability/E_2.html) - [E.3: Extension Theorem](http://theanalysisofdata.com/probability/E_3.html) - [E.4: Independence](http://theanalysisofdata.com/probability/E_4.html) - [E.5: Important Measures](http://theanalysisofdata.com/probability/E_5.html) - [E.6: Measurable Functions](http://theanalysisofdata.com/probability/E_6.html) - [E.7: Notes](http://theanalysisofdata.com/probability/E_7.html) - [F](http://theanalysisofdata.com/probability/3_12.html) - Integration - [F.1: Riemann Integral](http://theanalysisofdata.com/probability/F_1.html) - [F.2: Integration and Differentiation](http://theanalysisofdata.com/probability/F_2.html) - [F.3: Lebesgue Integral](http://theanalysisofdata.com/probability/F_3.html) - [F.4: Product Measures](http://theanalysisofdata.com/probability/F_4.html) - [F.5: Product Integration](http://theanalysisofdata.com/probability/F_5.html) - [F.6: Multivariate Extensions](http://theanalysisofdata.com/probability/F_6.html) - [F.7: Notes](http://theanalysisofdata.com/probability/F_7.html) \$ \\def\\P{\\mathsf{\\sf P}} \\def\\E{\\mathsf{\\sf E}} \\def\\Var{\\mathsf{\\sf Var}} \\def\\Cov{\\mathsf{\\sf Cov}} \\def\\std{\\mathsf{\\sf std}} \\def\\Cor{\\mathsf{\\sf Cor}} \\def\\R{\\mathbb{R}} \\def\\c{\\,\|\\,} \\def\\bb{\\boldsymbol} \\def\\diag{\\mathsf{\\sf diag}} \$ ## 3\.12. The Beta Distribution The beta RV \$\\text{Beta}(\\alpha,\\beta)\$, where \$\\alpha,\\beta \> 0\$, has the following pdf \\begin{align\*} f\_X(x) &= \\begin{cases} \\frac{1}{B(\\alpha,\\beta)} x^{\\alpha-1}(1-x)^{\\beta-1} & x\\in\[0,1\]\\\\ 0 & x\\not\\in\[0,1\] \\end{cases}, \\end{align\*} where \\begin{align\*} B(\\alpha,\\beta) = \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)} \\end{align\*} is the beta function. Example F.6.1 verifies that the pdf integrates to 1 and can be used to compute the expectation and other moments \\begin{align\*} \\E(X^k) &= \\frac{1}{B(\\alpha,\\beta)} \\int\_0^{\\infty} x^{\\alpha+k-1}(1-x)^{\\beta-1}\\,dx = \\frac{B(\\alpha+k,\\beta)}{B(\\alpha,\\beta)}\\\\ \\E(X) &= \\frac{B(\\alpha+1,\\beta)}{B(\\alpha,\\beta)} = \\frac{\\alpha}{\\alpha+\\beta}. \\end{align\*} If \$\\alpha=\\beta=1\$ the beta distribution reduces to the uniform distribution over \$\[0,1\]\$. For other values of \$\\alpha,\\beta\$, however, we get a different behavior. When \$\\alpha 1\$ and \$\\beta\>1\$ the pdf is unimodal, with a local maximum in \$(0,1)\$. If \$\\alpha=\\beta\$ the pdf is symmetric around 1/2 and if \$\\alpha\\neq \\beta\$ the pdf is asymmetric around 1/2. The R code below graphs pdf functions of the beta distribution. ``` x = seq(0, 1, length = 100) y1 = dbeta(x, 1/2, 1/2) y2 = dbeta(x, 1/2, 1) y3 = dbeta(x, 1/2, 2) y4 = dbeta(x, 1, 1/2) y5 = dbeta(x, 1, 1) y6 = dbeta(x, 1, 2) y7 = dbeta(x, 2, 1/2) y8 = dbeta(x, 2, 1) y9 = dbeta(x, 2, 2) D = data.frame(probability = c(y1, y2, y3, y4, y5, y6, y7, y8, y9)) D$x = x D$alpha[1:300] = "$\\alpha=1/2$" D$alpha[301:600] = "$\\alpha=1$" D$alpha[601:900] = "$\\alpha=2$" D$beta[1:100] = "$\\beta=1/2$" D$beta[101:200] = "$\\beta=1$" D$beta[201:300] = "$\\beta=2$" D$beta[301:400] = "$\\beta=1/2$" D$beta[401:500] = "$\\beta=1$" D$beta[501:600] = "$\\beta=2$" D$beta[601:700] = "$\\beta=1/2$" D$beta[701:800] = "$\\beta=1$" D$beta[801:900] = "$\\beta=2$" qplot(x, probability, main = "Beta pdf functions", data = D, geom = "area", facets = alpha ~ beta, xlab = "$x$", ylab = "$f_X(x)$", ) + scale_y_continuous(limits = c(0, 4)) ``` 

\$ \\def\\P{\\mathsf{\\sf P}} \\def\\E{\\mathsf{\\sf E}} \\def\\Var{\\mathsf{\\sf Var}} \\def\\Cov{\\mathsf{\\sf Cov}} \\def\\std{\\mathsf{\\sf std}} \\def\\Cor{\\mathsf{\\sf Cor}} \\def\\R{\\mathbb{R}} \\def\\c{\\,\|\\,} \\def\\bb{\\boldsymbol} \\def\\diag{\\mathsf{\\sf diag}} \$ The beta RV \$\\text{Beta}(\\alpha,\\beta)\$, where \$\\alpha,\\beta \> 0\$, has the following pdf \\begin{align\*} f\_X(x) &= \\begin{cases} \\frac{1}{B(\\alpha,\\beta)} x^{\\alpha-1}(1-x)^{\\beta-1} & x\\in\[0,1\]\\\\ 0 & x\\not\\in\[0,1\] \\end{cases}, \\end{align\*} where \\begin{align\*} B(\\alpha,\\beta) = \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)} \\end{align\*} is the beta function. Example F.6.1 verifies that the pdf integrates to 1 and can be used to compute the expectation and other moments \\begin{align\*} \\E(X^k) &= \\frac{1}{B(\\alpha,\\beta)} \\int\_0^{\\infty} x^{\\alpha+k-1}(1-x)^{\\beta-1}\\,dx = \\frac{B(\\alpha+k,\\beta)}{B(\\alpha,\\beta)}\\\\ \\E(X) &= \\frac{B(\\alpha+1,\\beta)}{B(\\alpha,\\beta)} = \\frac{\\alpha}{\\alpha+\\beta}. \\end{align\*} If \$\\alpha=\\beta=1\$ the beta distribution reduces to the uniform distribution over \$\[0,1\]\$. For other values of \$\\alpha,\\beta\$, however, we get a different behavior. When \$\\alpha 1\$ and \$\\beta\>1\$ the pdf is unimodal, with a local maximum in \$(0,1)\$. If \$\\alpha=\\beta\$ the pdf is symmetric around 1/2 and if \$\\alpha\\neq \\beta\$ the pdf is asymmetric around 1/2. The R code below graphs pdf functions of the beta distribution. ``` x = seq(0, 1, length = 100) y1 = dbeta(x, 1/2, 1/2) y2 = dbeta(x, 1/2, 1) y3 = dbeta(x, 1/2, 2) y4 = dbeta(x, 1, 1/2) y5 = dbeta(x, 1, 1) y6 = dbeta(x, 1, 2) y7 = dbeta(x, 2, 1/2) y8 = dbeta(x, 2, 1) y9 = dbeta(x, 2, 2) D = data.frame(probability = c(y1, y2, y3, y4, y5, y6, y7, y8, y9)) D$x = x D$alpha[1:300] = "$\\alpha=1/2$" D$alpha[301:600] = "$\\alpha=1$" D$alpha[601:900] = "$\\alpha=2$" D$beta[1:100] = "$\\beta=1/2$" D$beta[101:200] = "$\\beta=1$" D$beta[201:300] = "$\\beta=2$" D$beta[301:400] = "$\\beta=1/2$" D$beta[401:500] = "$\\beta=1$" D$beta[501:600] = "$\\beta=2$" D$beta[601:700] = "$\\beta=1/2$" D$beta[701:800] = "$\\beta=1$" D$beta[801:900] = "$\\beta=2$" qplot(x, probability, main = "Beta pdf functions", data = D, geom = "area", facets = alpha ~ beta, xlab = "$x$", ylab = "$f_X(x)$", ) + scale_y_continuous(limits = c(0, 4)) ```