Home page for accesible maths 7.2 Exponential Distribution:

{\rm Exp}(\beta)

7.2.2 The Gamma function 7.3 Gamma Distribution:

{\rm Gamma}(\alpha,\beta)

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7.2.3 Expectation and variance of ${\rm Exp}(\beta)$ rvs

The $r$ th moment of a general random variable $X$ is defined to be ${\rm E}(X^{r})$ . In the case of exponential random variables, we have that

$\displaystyle{\rm E}(X^{r})$	$\displaystyle=$	$\displaystyle\int_{-\infty}^{\infty}x^{r}f_{X}(x)\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}x^{r}\beta\exp(-\beta x)\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\beta\int_{0}^{\infty}x^{r}\exp(-\beta x)\,{\rm d}x.$

We will need to evaluate integrals of this form so many times that we create a lemma:

Lemma 7.9.

\int_{0}^{\infty}x^{\alpha-1}\exp(-\beta x)\,dx=\frac{\Gamma(\alpha)}{\beta^{% \alpha}}.

Proof.

Substituting $t=\beta x$ gives

$\displaystyle\int_{0}^{\infty}x^{\alpha-1}\exp(-\beta x)\,{\rm d}x$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}\left(\frac{t}{\beta}\right)^{\alpha-1}\exp(-t)% \,\frac{\,{\rm d}t}{\beta}$
	$\displaystyle=$	$\displaystyle\frac{1}{\beta^{\alpha}}\int_{0}^{\infty}t^{\alpha-1}\exp(-t)\,{% \rm d}t$
	$\displaystyle=$	$\displaystyle\frac{\Gamma(\alpha)}{\beta^{\alpha}}.$

∎

Using Lemma 7.9 with $\alpha=r+1$ we see that

$\displaystyle{\rm E}(X^{r})$	$\displaystyle=$	$\displaystyle\beta\int_{0}^{\infty}x^{r}e^{-\beta x}\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\beta\frac{\Gamma(r+1)}{\beta^{r+1}}$
	$\displaystyle=$	$\displaystyle\frac{\Gamma(r+1)}{\beta^{r}}.$

For integer values of $r$ , therefore ${\rm E}(X^{r})=r!/\beta^{r}$ .

In particular the expectation and variance of an exponential random variable are

	$\displaystyle{\rm E}(X)$	$\displaystyle=$	$\displaystyle\frac{1}{\beta}$
	$\displaystyle\ {\rm Var}(X)$	$\displaystyle=$	$\displaystyle\frac{2}{\beta^{2}}-\frac{1}{\beta^{2}}=\frac{1}{\beta^{2}}.$

Hence the expectation and standard deviation are the same. Note that the expectation decreases with $\beta$ ; $\beta$ is the rate at which events occur, so the higher the rate of events the shorter the expected waiting time to the next event.

Beware: Some computer packages use a different parameterisation of exponential random variables, in which an ${\rm Exp}(m)$ random variable has expected value $m$ …

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7.2.3 Expectation and variance of Exp⁢(β) rvs

Lemma 7.9.

Proof.

7.2.3 Expectation and variance of ${\rm Exp}(\beta)$ rvs