Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Beta distribution ▷ Moment-generating function

Theorem: Let $X$ be a positive random variable following a beta distribution:

\[\label{eq:beta} X \sim \mathrm{Bet}(\alpha, \beta) \; .\]

Then, the moment-generating function of $X$ is

\[\label{eq:beta-mgf} M_X(t) = 1 + \sum_{n=1}^{\infty} \left( \prod_{m=0}^{n-1} \frac{\alpha + m}{\alpha + \beta + m} \right) \frac{t^n}{n!} \; .\]

Proof: The probability density function of the beta distribution is

\[\label{eq:beta-pdf} f_X(x) = \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1}\]

and the moment-generating function is defined as

\[\label{eq:mgf-var} M_X(t) = \mathrm{E} \left[ e^{tX} \right] \; .\]

Using the expected value for continuous random variables, the moment-generating function of $X$ therefore is

\[\label{eq:beta-mgf-s1} \begin{split} M_X(t) &= \int_{0}^{1} \exp[tx] \cdot \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1} \, \mathrm{d}x \\ &= \frac{1}{\mathrm{B}(\alpha, \beta)} \int_{0}^{1} e^{tx} \, x^{\alpha-1} \, (1-x)^{\beta-1} \, \mathrm{d}x \; . \end{split}\]

With the relationship between beta function and gamma function

\[\label{eq:beta-gam-fct} \mathrm{B}(\alpha, \beta) = \frac{\Gamma(\alpha) \, \Gamma(\beta)}{\Gamma(\alpha+\beta)}\]

and the integral representation of the confluent hypergeometric function (Kummer’s function of the first kind)

\[\label{eq:con-hyp-geo-fct-int} {}_1 F_1(a,b,z) = \frac{\Gamma(b)}{\Gamma(a) \, \Gamma(b-a)} \int_{0}^{1} e^{zu} \, u^{a-1} \, (1-u)^{(b-a)-1} \, \mathrm{d}u \; ,\]

the moment-generating function can be written as

\[\label{eq:beta-mgf-s2} M_X(t) = {}_1 F_1(\alpha,\alpha+\beta,t) \; .\]

Note that the series equation for the confluent hypergeometric function (Kummer’s function of the first kind) is

\[\label{eq:con-hyp-geo-fct-ser} {}_1 F_1(a,b,z) = \sum_{n=0}^{\infty} \frac{a^{\overline{n}}}{b^{\overline{n}}} \, \frac{z^n}{n!}\]

where $m^{\overline{n}}$ is the rising factorial

\[\label{eq:fact-rise} m^{\overline{n}} = \prod_{i=0}^{n-1} (m+i) \; ,\]

so that the moment-generating function can be written as

\[\label{eq:beta-mgf-s3} M_X(t) = \sum_{n=0}^{\infty} \frac{\alpha^{\overline{n}}}{(\alpha+\beta)^{\overline{n}}} \, \frac{t^n}{n!} \; .\]

Applying the rising factorial equation \eqref{eq:fact-rise} and using $m^{\overline{0}} = x^0 = 0! = 1$, we finally have:

\[\label{eq:beta-mgf-s4} M_X(t) = 1 + \sum_{n=1}^{\infty} \left( \prod_{m=0}^{n-1} \frac{\alpha + m}{\alpha + \beta + m} \right) \frac{t^n}{n!} \; .\]
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Metadata: ID: P198 | shortcut: beta-mgf | author: JoramSoch | date: 2020-11-25, 06:55.