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Chapter 5: Problem 9

Let \(X_{1}, X_{2}, \ldots, X_{r}\) be independent exponential random variables with parameter \(\lambda .\) a. Find the moment-generating function of \(Y=X_{1}+X_{2}+\) \(\ldots+X_{r}\) b. What is the distribution of the random variable \(Y ?\)

Short Answer

Expert verified
a. MGF of Y is \(\left(\frac{\lambda}{\lambda - t}\right)^r\). b. Y is Gamma distributed: \(Y \sim \text{Gamma}(r, \lambda)\).

Step by step solution

01

Understand the Moment-Generating Function (MGF)

The moment-generating function (MGF) of a random variable gives us a way to find all moments of the random variable. For an exponential random variable \(X_i\) with parameter \(\lambda\), its MGF is \(M_{X_i}(t) = \frac{\lambda}{\lambda - t}\) for \(t < \lambda\).
02

Find the MGF of Y

Since \(Y = X_1 + X_2 + \ldots + X_r\) is a sum of independent random variables, the MGF of \(Y\), \(M_Y(t)\), is the product of the MGFs of the \(X_i\)'s. Therefore, \(M_Y(t) = \left(\frac{\lambda}{\lambda - t}\right)^r\).
03

Identify the Distribution

The MGF \(M_Y(t) = \left(\frac{\lambda}{\lambda - t}\right)^r\) is characteristic of a Gamma distribution with shape parameter \(r\) and rate parameter \(\lambda\). Gamma distributions have MGFs in the form of \(\left(\frac{\lambda}{\lambda - t}\right)^r\).
04

Conclusion about the Distribution of Y

With the shape parameter \(r\) and rate parameter \(\lambda\), \(Y\) is a Gamma distributed random variable, specifically, \(Y \sim \text{Gamma}(r, \lambda)\). If \(r\) is an integer, this is also known as an Erlang distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment-Generating Function
The moment-generating function (MGF) is a mathematical tool used to gain insights about the distribution of a random variable. It can provide us with all the moments of the random variable, which are expectations of powers of the variable.
For an exponential random variable with a rate parameter \( \lambda \), its moment-generating function \( M_{X}(t) \) is defined as \( \frac{\lambda}{\lambda - t} \) for \( t < \lambda \).
This function acts like a summary encapsulating the whole distribution of the variable, making it an extremely valuable tool in probability and statistics.
  • MGFs are especially useful in finding the distribution of sums of random variables.
  • They simplify the calculation of expected values and variances.
  • Can be used to approximate distributions when dealing with sums of large numbers of variables.
Gamma Distribution
The gamma distribution is a flexible distribution that generalizes the exponential distribution. It is characterized by two parameters: the shape parameter (often denoted by \( k \) or \( r \)) and the rate parameter \( \lambda \).
When dealing with sums of exponential random variables, the gamma distribution frequently appears.
  • If \( Y = X_1 + X_2 + \ldots + X_r \) where each \( X_i \) is an independent exponential variable, \( Y \) follows a gamma distribution.
  • The gamma distribution's MGF is \( \left( \frac{\lambda}{\lambda - t} \right)^r \).
Understanding gamma distributions makes it easier to work with processes involving intervals such as waiting times or workloads.
Independent Random Variables
Independent random variables are variables that do not affect each other's outcomes. In other words, the result of one random variable does not provide any information about another.
This is a crucial property when analyzing the sum of random variables.
  • Independence means the probability of one event occurring does not influence the probability of another.
  • When summing independent variables, the MGF of the resulting sum is the product of the individual MGFs.
  • For the given problem, each \( X_i \) being independent allows us to compute the MGF of \( Y \) directly from the MGFs of \( X_i \).
Ensuring you understand independence is key in understanding how processes unfold over time, especially in stochastic processes.
Erlang Distribution
The Erlang distribution is a special case of the gamma distribution where the shape parameter is an integer.
This distribution is often used in scenarios where events occur continuously and independently at a constant average rate.
  • If a gamma distribution's shape parameter \( r \) is an integer, it becomes an Erlang distribution: \( Y \sim \text{Erlang}(r, \lambda) \).
  • Commonly used in telecommunications and queueing theory to model waiting times.
  • MGF remains \( \left( \frac{\lambda}{\lambda - t} \right)^r \) reflecting the sum of \( r \) exponential variables.
Understanding the Erlang distribution is particularly valuable in engineering and operations research, where predicting timings is critical.

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Most popular questions from this chapter

A Web site uses ads to route visitors to one of four landing pages. The probabilities for each landing page are equal. Consider 20 independent visitors and let the random variables \(W, X, Y,\) and \(Z\) denote the number of visitors routed to each page. Calculate the following: a. \(P(W=5, X=5, Y=5, Z=5)\) b. \(P(W=5, X=5, Y=5, Z=5)\) c. \(P(W=7, X=7, Y=6 \mid Z=3)\) d. \(P(W=7, X=7, Y=3 \mid Z=3)\) e. \(P(W \leq 2)\) f. \(E(W)\) g. \(P(W=5, X=5)\) h. \(P(W=5 \mid X=5)\)

Determine the value of \(c\) such that the function \(f(x, y)=c x^{2} y\) for \(02.1

A continuous random variable \(X\) has the following probability distribution: $$ f(x)=4 x e^{-2 x}, \quad x>0 $$ a. Find the moment-generating function for \(X\). b. Find the mean and variance of \(X\).

The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of two integrated layers. The layers are normally distributed with means of 0.5 and 1 millimeters, respectively. The standard deviations of layer thickness are 0.1 and 0.2 millimeters, respectively. The correlation between layers is \(0.7 .\) a. Determine the mean and variance of the total thickness of the two layers. b. What is the probability that the total thickness is less than I millimeter? c. Let \(X_{1}\) and \(X_{2}\) denote the thickness of layers 1 and 2 , respectively. A measure of performance of the membrane is a function of \(2 X_{1}+3 X_{2}\) of the thickness. Determine the mean and variance of this performance measure.

A small company is to decide what investments to use for cash generated from operations. Each investment has a mean and standard deviation associated with a percentage return. The first security has a mean return of \(5 \%\) with a standard deviation of \(2 \%,\) and the second security provides the same mean of \(5 \%\) with a standard deviation of \(4 \%\). The securities have a correlation of \(-0.5 .\) If the company invests \(\$ 2\) million with half in each security, what are the mean and standard deviation of the percentage return? Compare the standard deviation of this strategy to one that invests the \(\$ 2\) million into the first security only.
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