# Exponential Distribution | MGF | PDF | Mean | Variance

The Exponential Distribution is one of the continuous distribution used to measure time the expected time for an event to occur.

A continuous random variable $X$ is said to have an exponential distribution with parameter $\theta$ if its probability denisity function is given by

 \begin{align*} f(x)&= \begin{cases} \theta e^{-\theta x}, & x>0;\theta>0 \\ 0, & Otherwise. \end{cases} \end{align*}

In notation, it can be written as $X\sim \exp(\theta)$.

Another form of exponential distribution is

 \begin{align*} f(x)&= \begin{cases} \frac{1}{\theta} e^{-\frac{x}{\theta}}, & x>0;\theta>0 \\ 0, & Otherwise. \end{cases} \end{align*}

In notation, it can be written as $X\sim \exp(1/\theta)$.

## Distribution Function

The distribution function of exponential distribution is $F(x) = 1-e^{-\theta x}$.

### Proof

The distribution function of exponential distribution is

 $$\begin{eqnarray*} F(x) &=& P(X\leq x) \\ &=& \int_0^x f(x)\;dx\\ &=& \theta \int_0^x e^{-\theta x}\;dx\\ &=& \theta \bigg[-\frac{e^{-\theta x}}{\theta}\bigg]_0^x \\ &=& 1-e^{-\theta x}. \end{eqnarray*}$$

## Mean and Variance of Exponential Distribution

Let $X\sim \exp(\theta)$. Then the mean and variance of $X$ are $\frac{1}{\theta}$ and $\frac{1}{\theta^2}$ respectively.

### Mean and Variance Proof

The mean of exponential distribution is

 $$\begin{eqnarray*} \text{mean = }\mu_1^\prime &=& E(X) \\ &=& \int_0^\infty x\theta e^{-\theta x}\; dx\\ &=& \theta \int_0^\infty x^{2-1}e^{-\theta x}\; dx\\ &=& \theta \frac{\Gamma(2)}{\theta^2}\;\quad (\text{Using }\int_0^\infty x^{n-1}e^{-\theta x}\; dx = \frac{\Gamma(n)}{\theta^n} )\\ &=& \frac{1}{\theta} \end{eqnarray*}$$

To find the variance, we need to find $E(X^2)$.

 $$\begin{eqnarray*} \mu_2^\prime&= &E(X^2)\\ &=& \int_0^\infty x^2\theta e^{-\theta x}\; dx\\ &=& \theta \int_0^\infty x^{3-1}e^{-\theta x}\; dx\\ &=& \theta \frac{\Gamma(3)}{\theta^3}\;\quad (\text{Using }\int_0^\infty x^{n-1}e^{-\theta x}\; dx = \frac{\Gamma(n)}{\theta^n} )\\ &=& \frac{2}{\theta^2} \end{eqnarray*}$$

Hence, the variance of exponential distribution is

 $$\begin{eqnarray*} \text{Variance = } \mu_2&=&\mu_2^\prime-(\mu_1^\prime)^2\\ &=&\frac{2}{\theta^2}-\bigg(\frac{1}{\theta}\bigg)^2\\ &=&\frac{1}{\theta^2}. \end{eqnarray*}$$

## Raw Moments of Exponential Distribution

Let $X\sim \exp(\theta)$. The $r^{th}$ raw moment of exponential distribution is $\mu_r^\prime = \frac{r!}{\theta^r}$.

### Raw Moments Proof

The $r^{th}$ raw moment of exponential distribution is

 $$\begin{eqnarray*} \mu_r^\prime &=& E(X^r) \\ &=& \int_0^\infty x^r\theta e^{-\theta x}\; dx\\ &=& \theta \int_0^\infty x^{(r+1)-1}e^{-\theta x}\; dx\\ &=& \theta \frac{\Gamma(r+1)}{\theta^{r+1}}\;\quad (\text{Using }\int_0^\infty x^{n-1}e^{-\theta x}\; dx = \frac{\Gamma(n)}{\theta^n} )\\ &=& \frac{r!}{\theta^r}\;\quad (\because \Gamma(n) = (n-1)!) \end{eqnarray*}$$

## Moments of Generating Function (M.G.F.) of exponential Distribution

Let $X\sim\exp(\theta)$. The moment generating function of exponential distribution is $M_X(t)= \big(1-\frac{t}{\theta}\big)^{-1}$.

### M.G.F. of Exponential Distribution Proof

The moment generating function of $X$ is
 $$\begin{eqnarray*} M_X(t) &=& E(e^{tX}) \\ &=& \int_0^\infty e^{tx}\theta e^{-\theta x}\; dx\\ &=& \theta \int_0^\infty e^{-(\theta-t) x}\; dx\\ &=& \theta \bigg[-\frac{e^{-(\theta-t) x}}{\theta-t}\bigg]_0^\infty \; dx\\ &=& \frac{\theta }{\theta-t}\bigg[e^{-(\theta-t) x}\bigg]_0^\infty \; dx\\ &=& \frac{\theta }{\theta-t}, \text{ (if t<\theta})\\ &=& \big(1-\frac{t}{\theta}\big)^{-1}. \end{eqnarray*}$$

## Memoryless Property of Exponential Distribution

For an exponential random variable $X$ with parameter $\theta$ and for $s,t\geq 0$,

 $$\begin{equation*} P(X>s+t|X>s) = P(X>t). \end{equation*}$$

#### Proof

Let $X\sim exp(\theta)$. Then the distribution function of $X$ is
$F(x)=1-e^{-\theta x}$.

Then

 $$\begin{eqnarray*} P(X>s+t | X>s) &=& \frac{P(X> s+t, X> s)}{P(X>s)}\\ &=&\frac{P(X>s+t)}{P(X>s)}\\ &=&\frac{1-P(X\leq s+t)}{1-P(X\leq s)}\\ &=&\frac{1-F(s+t)}{1-F(s)}\\ &=&\frac{e^{-\theta (s+t)}}{e^{-\theta s}}\\ &=& e^{-\theta t}\\ &=& 1-F(t)\\ &=&P(X>t). \end{eqnarray*}$$

The above property of an exponential distribution is known as memoryless property.

Exponential distribution is the only continuous distribution which have the memoryless property.

## Reference

Refer Exponential Distribution Calculator to find the probability density and cumulative probabilities for Exponential distribution with parameter $\theta$ and examples.

Exponential Distribution Calculator

Let me know in the comments if you have any questions on Exponential Distribution ,M.G.F. and P.D.F and your thought on this article.