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Chapter 2: Problem 64

Show that the sum of independent identically distributed exponential random variables has a gamma distribution.

Short Answer

Expert verified
The sum of n independent identically distributed exponential random variables, denoted by \(Y = X_1 + X_2 + \dots + X_n\), with a rate parameter \(\lambda\) follows a gamma distribution. Its probability density function is given by \(f_Y(y) = \frac{\lambda^n}{(n-1)!}y^{n-1}e^{-\lambda y}\) for \(x \geq 0\), and \(Y \sim \Gamma(n, \frac{1}{\lambda})\).

Step by step solution

01

Define exponential random variables and their pdf

An exponential random variable, denoted by X, is a continuous random variable with the pdf given by: \(f_X(x) = \lambda e^{-\lambda x}\), for \(x \geq 0\) where \(\lambda > 0\) is the rate parameter.
02

Define gamma distribution and its pdf

A random variable Y follows a gamma distribution, denoted by \(Y \sim \Gamma(k, \theta)\), if its pdf is given by: \[f_Y(y) = \frac{1}{\Gamma(k)\theta^k} y^{k-1} e^{-\frac{y}{\theta}}\], for \(y \geq 0\) where \(k > 0\) is the shape parameter, \(\theta > 0\) is the scale parameter, and \(\Gamma(k)\) is the gamma function.
03

Define the sum of independent exponential random variables

Let \(X_1, X_2, \dots, X_n\) be n independent identically distributed exponential random variables with the same rate parameter \(\lambda\). We want to find the distribution of the sum of these random variables, Y, where \(Y = X_1 + X_2 + \dots + X_n\).
04

Find the pdf of the sum of independent exponential random variables using convolution

Since these random variables are independent, we can find the pdf of Y (denoted as \(f_Y(y)\)) by using the convolution theorem. For n = 2, the convolution can be expressed as: \(f_Y(y) = \int_{-\infty}^{\infty} f_{X_1}(x) f_{X_2}(y-x) dx = \int_0^y \lambda e^{-\lambda x} \lambda e^{-\lambda(y-x)} dx = \lambda^2 ye^{-\lambda y}\), for \(0 \leq x \leq y\). Next, we will extend this result to n random variables. It can be shown that the pdf for the sum of n independent exponential random variables is: \(f_Y(y) = \frac{\lambda^n}{(n-1)!}y^{n-1}e^{-\lambda y}\), for \(x \geq 0\).
05

Show that the result is a gamma distribution

Comparing the result from step 4 with the gamma distribution pdf in step 2, we see that the pdf of the sum of independent exponential random variables follows a gamma distribution with shape parameter \(k=n\) and scale parameter \(\theta=\frac{1}{\lambda}\): \(f_Y(y) = \frac{1}{\Gamma(n)(\frac{1}{\lambda})^n}y^{n-1}e^{-\frac{y}{\frac{1}{\lambda}}}\). Therefore, \(Y \sim \Gamma(n, \frac{1}{\lambda})\). This shows that the sum of independent identically distributed exponential random variables has a gamma distribution.

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Most popular questions from this chapter

Let \(c\) be a constant. Show that (a) \(\operatorname{Var}(c X)=c^{2} \operatorname{Var}(X)\) (b) \(\operatorname{Var}(c+X)=\operatorname{Var}(X)\).

Let \(X\) and \(Y\) each take on either the value 1 or \(-1\). Let $$ \begin{aligned} p(1,1) &=P\\{X=1, Y=1\\} \\ p(1,-1) &=P[X=1, Y=-1\\} \\ p(-1,1) &=P[X=-1, Y=1\\} \\ p(-1,-1) &=P\\{X=-1, Y=-1\\} \end{aligned} $$ Suppose that \(E[X]=E[Y]=0\). Show that (a) \(p(1,1)=p(-1,-1) ;\) (b) \(p(1,-1)=p(-1,1)\). Let \(p=2 p(1,1) .\) Find (c) \(\operatorname{Var}(X)\); (d) \(\operatorname{Var}(Y)\) (e) \(\operatorname{Cov}(X, Y)\).

Suppose that an experiment can result in one of \(r\) possible outcomes, the ith outcome having probability \(p_{i}, i=1, \ldots, r, \sum_{i=1}^{r} p_{i}=1 .\) If \(n\) of these experiments are performed, and if the outcome of any one of the \(n\) does not affect the outcome of the other \(n-1\) experiments, then show that the probability that the first outcome appears \(x_{1}\) times, the second \(x_{2}\) times, and the \(r\) th \(x_{r}\) times is $$ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \quad \text { when } x_{1}+x_{2}+\cdots+x_{r}=n $$ This is known as the multinomial distribution.

Suppose five fair coins are tossed. Let \(E\) be the event that all coins land heads. Define the random variable \(I_{E}\) $$ I_{E}=\left\\{\begin{array}{ll} 1, & \text { if } E \text { occurs } \\ 0, & \text { if } E^{c} \text { occurs } \end{array}\right. $$ For what outcomes in the original sample space does \(I_{E}\) equal 1? What is \(P\left[I_{E}=1\right\\}\) ?

An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?
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