Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Exponential distribution ▷ Moment-generating function

Theorem: Let $X$ be a random variable following an exponential distribution:

$\label{eq:exp} X \sim \mathrm{Exp}(\lambda) \; .$

Then, the moment generating function of $X$ is

$\label{eq:exp-mgf} M_X(t) = \frac{\lambda}{\lambda - t}$

which is well-defined for $t < \lambda$.

Proof: Suppose $X$ follows an exponential distribution with rate $\lambda$; that is, $X\sim \mathrm{Exp}(\lambda)$. Then, the probability density function is given by

$\label{eq:exp-pdf} f_X(x) = \lambda e^{-\lambda x}$

and the moment-generating function is defined as

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

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

$\label{eq:exp-mgf-s1} \begin{split} M_X(t) &= \int_0^{\infty} e^{tx} \cdot f_X(x) dx\\ &= \int_0^{\infty} e^{tx}\cdot \lambda e^{-\lambda x} dx\\ &= \int_0^{\infty} \lambda e^{x(t-\lambda)} dx\\ &= \frac{\lambda}{t-\lambda} e^{x(t-\lambda)} \Big|_{x = 0}^{x = \infty}\\ &= \lim_{x\rightarrow \infty} \left[ \frac{\lambda}{t-\lambda} e^{x(t-\lambda)} - \frac{\lambda}{t-\lambda}\right]\\ &= \frac{\lambda}{t-\lambda} \left[ \lim_{x\rightarrow \infty} e^{x(t-\lambda)} -1 \right] \; . \end{split}$

Note that $t$ cannot be equal to $\lambda$, else $M_X(t)$ is undefined. Further, if $t > \lambda$, then $\lim_{x\rightarrow \infty} e^{x(t-\lambda)} = \infty$, which implies that $M_X(t)$ diverges for $t \geq \lambda$. So, we must restrict the domain of $M_X(t)$ to $t<\lambda$. Assuming this, we can further simplify \eqref{eq:exp-mgf-s1}:

$\label{eq:exp-mgf-s2} \begin{split} M_X(t) &= \frac{\lambda}{t-\lambda} \left[ \lim_{x\rightarrow \infty} e^{x(t-\lambda)} -1 \right] \\ &= \frac{\lambda}{t-\lambda} \left[ 0 - 1 \right] \\ &= \frac{\lambda}{\lambda - t} \; . \end{split}$

This completes the proof of \eqref{eq:exp-mgf}.

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Metadata: ID: P403 | shortcut: exp-mgf | author: tomfaulkenberry | date: 2023-04-19, 12:00.