Exponential Distribution

A continuous Random Variable is of exponential distribution if it has

• Parameters
  • $\lambda >0$
• PDF
  • $f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x},& x\ge 0,\\ 0,& x<0\end{array}$
  • The important part is $e^{-\lambda x}$; $\lambda$ is just a constant to make if integral to 1
• CDF
  • $1-e^{-\lambda x}$
• Mean
  • $\frac{1}{\lambda }$
• Variance
  • $\frac{1}{{\lambda }^2}$
• MGF
  • $\lambda /(\lambda -t),t<\lambda$
• CF
  • $\lambda /(\lambda -it)$

By the uniqueness of MGF, exponential distribution is Gamma Distribution with parameter $(1,\lambda )$.

Exponential Distribution and Poisson Distribution

Exponential distribution and Poisson Distribution are similar in many ways. Actually, the waiting times for poisson distribution is an exponential distribution with parameter $\lambda$. Actually, let $Y$ be the waiting time, let $\lambda$ be the average number of arrivals per time. For $t\ge 0$, let $X\sim \mathrm{Poisson}(\lambda t)$. We know that

$$P(Y>t)=P(X=0)=e^{-\lambda t}\frac{(\lambda t{)}^0}{0!}=e^{-\lambda t}.$$

Then $P(Y\le t)=1-e^{-\lambda t}$, i.e., $Y\sim \exp (\lambda )$.

Exponential Distribution and Geometric Distribution

Exponential distribution can be viewed as the “limit”, or a continuous version, of a Geometric Distribution:

$$F_{\mathrm{Exp}}(n\delta )=1-e^{-\lambda n\delta }=1-(e^{-\lambda \delta }{)}^n=F_{\mathrm{Geo}}(n),$$

where the geometric distribution has a parameter $q=e^{-\lambda \delta }$. Intuitively, the exponential distribution corresponds to a limit of a situation where every $\delta$ time units, we toss a coin whose success probability is $p=1-e^{-\lambda \delta }\approx \lambda \delta$, and let $X$ be the time elapsed until the first success.

Memoryless

We say a nonnegative Random Variable $X$ is memoryless, if for $n,k\ge 0$

$$P(X\ge n+k|X\ge n)=P(X\ge k).$$

An exponential random variable is memoryless because

$$P(X\ge n+k|X\ge n)=\frac{e^{-\lambda (n+k)}}{e^{-\lambda n}}=e^{-\lambda k}=P(x\ge k).$$