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| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2020-11-25 06:55:00 |
| 9 | + |
| 10 | +title: "Moment-generating function of the beta distribution" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Univariate continuous distributions" |
| 13 | +topic: "Beta distribution" |
| 14 | +theorem: "Moment-generating function" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Wikipedia" |
| 18 | + year: 2020 |
| 19 | + title: "Beta distribution" |
| 20 | + in: "Wikipedia, the free encyclopedia" |
| 21 | + pages: "retrieved on 2020-11-25" |
| 22 | + url: "https://en.wikipedia.org/wiki/Beta_distribution#Moment_generating_function" |
| 23 | + - authors: "Wikipedia" |
| 24 | + year: 2020 |
| 25 | + title: "Confluent hypergeometric function" |
| 26 | + in: "Wikipedia, the free encyclopedia" |
| 27 | + pages: "retrieved on 2020-11-25" |
| 28 | + url: "https://en.wikipedia.org/wiki/Confluent_hypergeometric_function#Kummer's_equation" |
| 29 | + |
| 30 | +proof_id: "P198" |
| 31 | +shortcut: "beta-mgf" |
| 32 | +username: "JoramSoch" |
| 33 | +--- |
| 34 | + |
| 35 | + |
| 36 | +**Theorem:** Let $X$ be a positive [random variable](/D/rvar) following a [beta distribution](/D/gam): |
| 37 | + |
| 38 | +$$ \label{eq:beta} |
| 39 | +X \sim \mathrm{Bet}(\alpha, \beta) \; . |
| 40 | +$$ |
| 41 | + |
| 42 | +Then, the [moment-generating function](/D/mgf) of $X$ is |
| 43 | + |
| 44 | +$$ \label{eq:beta-mgf} |
| 45 | +M_X(t) = 1 + \sum_{n=1}^{\infty} \left( \prod_{m=0}^{n-1} \frac{\alpha + m}{\alpha + \beta + m} \right) \frac{t^n}{n!} \; . |
| 46 | +$$ |
| 47 | + |
| 48 | + |
| 49 | +**Proof:** The [probability density function of the beta distribution](/P/beta-pdf) is |
| 50 | + |
| 51 | +$$ \label{eq:beta-pdf} |
| 52 | +f_X(x) = \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1} |
| 53 | +$$ |
| 54 | + |
| 55 | +and the [moment-generating function](/D/mgf) is defined as |
| 56 | + |
| 57 | +$$ \label{eq:mgf-var} |
| 58 | +M_X(t) = \mathrm{E} \left[ e^{tX} \right] \; . |
| 59 | +$$ |
| 60 | + |
| 61 | +Using the [expected value for continuous random variables](/D/mean), the moment-generating function of $X$ therefore is |
| 62 | + |
| 63 | +$$ \label{eq:beta-mgf-s1} |
| 64 | +\begin{split} |
| 65 | +M_X(t) &= \int_{0}^{1} \exp[tx] \cdot \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1} \, \mathrm{d}x \\ |
| 66 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \int_{0}^{1} e^{tx} \, x^{\alpha-1} \, (1-x)^{\beta-1} \, \mathrm{d}x \; . |
| 67 | +\end{split} |
| 68 | +$$ |
| 69 | + |
| 70 | +With the relationship between beta function and gamma function |
| 71 | + |
| 72 | +$$ \label{eq:beta-gam-fct} |
| 73 | +\mathrm{B}(\alpha, \beta) = \frac{\Gamma(\alpha) \, \Gamma(\beta)}{\Gamma(\alpha+\beta)} |
| 74 | +$$ |
| 75 | + |
| 76 | +and the integral representation of the confluent hypergeometric function (Kummer's function of the first kind) |
| 77 | + |
| 78 | +$$ \label{eq:con-hyp-geo-fct-int} |
| 79 | +{}_1 F_1(a,b,z) = \frac{\Gamma(b)}{\Gamma(a) \, \Gamma(b-a)} \int_{0}^{1} e^{zu} \, u^{a-1} \, (1-u)^{(b-a)-1} \, \mathrm{d}u \; , |
| 80 | +$$ |
| 81 | + |
| 82 | +the moment-generating function can be written as |
| 83 | + |
| 84 | +$$ \label{eq:beta-mgf-s2} |
| 85 | +M_X(t) = {}_1 F_1(\alpha,\alpha+\beta,t) \; . |
| 86 | +$$ |
| 87 | + |
| 88 | +Note that the series equation for the confluent hypergeometric function (Kummer's function of the first kind) is |
| 89 | + |
| 90 | +$$ \label{eq:con-hyp-geo-fct-ser} |
| 91 | +{}_1 F_1(a,b,z) = \sum_{n=0}^{\infty} \frac{a^{\overline{n}}}{b^{\overline{n}}} \, \frac{z^n}{n!} |
| 92 | +$$ |
| 93 | + |
| 94 | +where $m^{\overline{n}}$ is the rising factorial |
| 95 | + |
| 96 | +$$ \label{eq:fact-rise} |
| 97 | +m^{\overline{n}} = \prod_{i=0}^{n-1} (m+i) \; , |
| 98 | +$$ |
| 99 | + |
| 100 | +so that the moment-generating function can be written as |
| 101 | + |
| 102 | +$$ \label{eq:beta-mgf-s3} |
| 103 | +M_X(t) = \sum_{n=0}^{\infty} \frac{\alpha^{\overline{n}}}{(\alpha+\beta)^{\overline{n}}} \, \frac{t^n}{n!} \; . |
| 104 | +$$ |
| 105 | + |
| 106 | +Applying the rising factorial equation \eqref{eq:fact-rise} and using $m^{\overline{0}} = x^0 = 0! = 1$, we finally have: |
| 107 | + |
| 108 | +$$ \label{eq:beta-mgf-s4} |
| 109 | +M_X(t) = 1 + \sum_{n=1}^{\infty} \left( \prod_{m=0}^{n-1} \frac{\alpha + m}{\alpha + \beta + m} \right) \frac{t^n}{n!} \; . |
| 110 | +$$ |
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