# 15.2 - Exponential Properties

15.2 - Exponential Properties

Here, we present and prove four key properties of an exponential random variable.

## Theorem

The exponential probability density function:

$$f(x)=\dfrac{1}{\theta} e^{-x/\theta}$$

for $$x\ge 0$$ and $$\theta>0$$ is a valid probability density function.

## Theorem

The moment generating function of an exponential random variable $$X$$ with parameter $$\theta$$ is:

$$M(t)=\dfrac{1}{1-\theta t}$$

for $$t<\frac{1}{\theta}$$.

### Proof

$$M(t)=E(e^{tX})=\int_0^\infty e^{tx} \left(\dfrac{1}{\theta}\right) e^{-x/\theta} dx$$

Simplifying and rewriting the integral as a limit, we have:

$$M(t)=\dfrac{1}{\theta}\lim\limits_{b \to \infty} \int_0^b e^{x(t-1/\theta)} dx$$

Integrating, we have:

$$M(t)=\dfrac{1}{\theta}\lim\limits_{b \to \infty} \left[ \dfrac{1}{t-1/\theta} e^{x(t-1/\theta)} \right]^{x=b}_{x=0}$$

Evaluating at $$x=0$$ and $$x=b$$, we have:

$$M(t)=\dfrac{1}{\theta}\lim\limits_{b \to \infty} \left[ \dfrac{1}{t-1/\theta} e^{b(t-1/\theta)} - \dfrac{1}{t-1/\theta} \right]=\dfrac{1}{\theta}\lim\limits_{b \to \infty} \left\{ \left(\dfrac{1}{t-1/\theta}\right) e^{b(t-1/\theta)} \right\}-\dfrac{1}{t-1/\theta}$$

Now, the limit approaches 0 provided $$t-\frac{1}{\theta}<0$$, that is, provided $$t<\frac{1}{\theta}$$, and so we have:

$$M(t)=\dfrac{1}{\theta} \left(0-\dfrac{1}{t-1/\theta}\right)$$

Simplifying more:

$$M(t)=\dfrac{1}{\theta} \left(-\dfrac{1}{\dfrac{\theta t-1}{\theta}}\right)=\dfrac{1}{\theta}\left(-\dfrac{\theta}{\theta t-1}\right)=-\dfrac{1}{\theta t-1}$$

and finally:

$$M(t)=\dfrac{1}{1-\theta t}$$

provided $$t<\frac{1}{\theta}$$, as was to be proved.

## Theorem

The mean of an exponential random variable $$X$$ with parameter $$\theta$$ is:

$$\mu=E(X)=\theta$$

## Theorem

The variance of an exponential random variable $$X$$ with parameter $$\theta$$ is:

$$\sigma^2=Var(X)=\theta^2$$

### Proof

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