# 4.7: Other Continuous Distributions

Exponential Distribution

Definition$$\PageIndex{1}$$

A random variable $$X$$ has a exponential distribution with parameter $$\lambda>0$$, write $$X$$ ~exponential$$(\lambda)$$, if $$X$$ has pdf given by
$$f(x) = \left\{\begin{array}{l l} \lambda e^{-\lambda x}, & \text{for}\ x\geq 0, \\ 0 & \text{otherwise.} \end{array}\right.\notag$$

A typical application of the exponential distribution is to model waiting times or lifetimes.  For example, each of the following gives an application of the exponential distribution.
\begin{itemize}
\item $$X=$$ lifetime of a radioactive particle
\item $$X=$$ how long you have to wait for an accident to occur at a given intersection
\item $$X=$$ length of interval between consecutive occurrences of Poisson distributed events
\end{itemize}

The parameter $$\lambda$$ is referred to as the rate parameter, it represents how quickly events occur.  For example, in the first case above where $$X$$ denotes the lifetime of a radioactive particle, $$\lambda$$ would give the rate at which such particles decay.

Properties of the Exponential Distribution:

\begin{enumerate}
\item The cdf of $$X$$ is given by
$$F(x) = \left\{\begin{array}{l l} 0 & \text{for}\ x< 0, \\ 1- e^{-\lambda x}, & \text{for}\ x\geq 0. \\ \end{array}\right.\notag$$
\item For any $$0<p<1$$, $$\displaystyle{\pi_p = \frac{-\ln(p)}{\lambda}}$$
\item $$\displaystyle{\expec{X} = \frac{1}{\lambda}}$$
\item $$\displaystyle{\var{X} = \frac{1}{\lambda^2}}$$
\item $$X$$ satisfies the Memoryless Property, i.e., $$P(X>t+s\ |\ X>s) = P(X>t)$$, for any $$t,s \geq0$$.
\end{enumerate}

\noindent \emph{\textbf{Proof:}}  We prove Properties \#1 \& \#3, the others are left as an exercise.
\newline
\#1) \begin{itemize}
\item For $x<0$: $F(x) = \int^{x}_{-\infty} f(t) dt = \int^x_{-\infty} 0 dt = 0$
\item For $x\geq 0$: $F(x) = \int^x_{-\infty} f(t) dt = \int^x_0 \lambda e^{-\lambda t} dt = -e^{-\lambda t}\Big|^x_0 = -e^{-\lambda t} - (-e^0) = 1-e^{-\lambda t}$
\end{itemize}
\#3)
\begin{itemize}
\item $\expec{X} = \int^{\infty}_{-\infty} x\cdot f(x) dx = \int^{\infty}_0 x\cdot \lambda e^{-\lambda x} dx = -x\cdot e^{-\lambda x}\big|^{\infty}_0 + \int^{\infty}_0 e^{-\lambda x} dx = 0 + \frac{-e^{-\lambda x}}{\lambda}\big|^{\infty}_0 = \frac{1}{\lambda}$
\end{itemize}
\newpage
\subsection{Gamma Distribution}
\begin{framed}
\begin{defn}
A random variable $X$ has a \define{gamma distribution} with parameters $\alpha, \lambda>0$, write $X\sim\text{gamma}(\alpha, \lambda)$, if $X$ has pdf given by
$$f(x) = \left\{\begin{array}{l l} \displaystyle{\frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}}, & \text{for}\ x\geq 0, \\ 0 & \text{otherwise,} \end{array}\right.$$
where $\Gamma$ is a function (referred to as the \define{gamma function}) given by the following integral:
$$\Gamma(\alpha) = \int^{\infty}_0 t^{\alpha-1}e^{-t}dt.$$
\end{defn}
\end{framed}
Note that the gamma function, $\Gamma(\alpha)$, ensures that the gamma pdf is valid, i.e., that it integrates to $1$.
\vskip0.25in
\emph{Show: } $\int^{\infty}_0 \frac{\lambda^\alpha}{\tau(\alpha)}x^{\alpha-1}e^{-\lambda x} dx = 1$
\newline
In the integral, we can make the substitution: $u = \lambda x \rightarrow du = \lambda dx$. Therefore, we have $\int^{\infty}_0 \frac{\lambda^\alpha}{\tau(\alpha)}x^{\alpha-1}e^{-\lambda x} dx =\\ \int^{\infty}_0 \frac{\lambda \lambda^{\alpha-1}}{\tau(\alpha)} x^{\alpha-1}e^{-\lambda x} dx = \frac{1}{\tau(\alpha)}\int^{\infty}_0 u^{\alpha-1}e^{-u} du = \frac{1}{\tau(\alpha)}\tau(\alpha) = 1$
\vskip0.5in

\begin{framed}
\noindent \textbf{Notes about the Gamma Distribution:}
\begin{enumerate}
\item If $\alpha = 1$, then the corresponding gamma distribution is given by the exponential distribution, i.e., $\text{gamma}(1,\lambda) = \text{exponential}(\lambda)$.  This is left as an exercise.
\item The parameter $\alpha$ is referred to as the \define{shape parameter}, and $\lambda$ is the \define{rate parameter} (or scale parameter).
\item A closed form doesn't exist for the cdf of a gamma distribution, and it's not commonly built into graphing calculators.\frownie
\end{enumerate}
\end{framed}
A typical application of the gamma distribution is to model the time it takes for a given number of events to occur.  For example, each of the following gives an application of the gamma distribution.
\begin{itemize}
\item $X=$ lifetime of 5 radioactive particles
\item $X=$ how long you have to wait for 3 accidents to occur at a given intersection
\end{itemize}
In these examples, the parameter $\lambda$ represents the rate at which the event occurs, and the parameter $\alpha$ is the number of events desired.  So, in the first example, $\alpha=5$ and $\lambda$ represents the rate at which particles decay.
\newpage