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.
