Expectation of Gamma Distribution
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Theorem
Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
The expectation of $X$ is given by:
- $\expect X = \dfrac \alpha \beta$
Proof 1
From the definition of the Gamma distribution, $X$ has probability density function:
- $\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From the definition of the expected value of a continuous random variable:
- $\ds \expect X = \int_0^\infty x \map {f_X} x \rd x$
So:
\(\ds \expect X\) | \(=\) | \(\ds \frac {\beta^\alpha} {\map \Gamma \alpha} \int_0^\infty x^\alpha e^{-\beta x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\beta^\alpha} {\map \Gamma \alpha} \int_0^\infty \left({\frac t \beta}\right)^\alpha e^{-t} \frac {\rd t} \beta\) | substituting $t = \beta x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\beta^\alpha} {\beta^{\alpha + 1} \map \Gamma \alpha} \int_0^\infty t^\alpha e^{-t} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\alpha + 1} } {\beta \map \Gamma \alpha}\) | Definition of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\alpha \map \Gamma \alpha} {\beta \map \Gamma \alpha}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \alpha \beta\) |
$\blacksquare$
Proof 2
By Moment Generating Function of Gamma Distribution, the moment generating function of $X$ is given by:
- $\map {M_X} t = \paren {1 - \dfrac t \beta}^{-\alpha}$
for $t < \beta$.
From Moment in terms of Moment Generating Function:
- $\expect X = \map { {M_X}'} 0$
From Moment Generating Function of Gamma Distribution: First Moment:
- $\map { {M_X}'} t = \dfrac {\beta^\alpha \alpha} {\paren {\beta - t}^{\alpha + 1} }$
Hence setting $t = 0$:
\(\ds \map { {M_X}'} 0\) | \(=\) | \(\ds \frac {\beta^\alpha \alpha} {\paren {\beta - 0}^{\alpha + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\beta^\alpha \alpha} {\beta^{\alpha + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \alpha \beta\) |
$\blacksquare$
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions