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4.7: Other Continuous Distributions

  • Page ID
    4380
  • 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