Suppose that the waiting time for the first customer to

Chapter 4, Problem 136E

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QUESTION:

Suppose that the waiting time for the first customer to enter a retail shop after . is a random variable  with an exponential density function given by

                                \(f(y)=\left\{\begin{array}{ll}

\left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0, \\

0, & \text { elsewhere. }

\end{array}\right.

\)

a Find the moment-generating function for .
b Use the answer from part (a) to find \(E(Y)\) and \(V(Y)\).

Equation Transcription:

Text Transcription:

f(y)=

(1 over theta)e^-y/theta, y>0

0, elsewhere.

E(Y)

V(Y)

Questions & Answers

QUESTION:

Suppose that the waiting time for the first customer to enter a retail shop after . is a random variable  with an exponential density function given by

                                \(f(y)=\left\{\begin{array}{ll}

\left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0, \\

0, & \text { elsewhere. }

\end{array}\right.

\)

a Find the moment-generating function for .
b Use the answer from part (a) to find \(E(Y)\) and \(V(Y)\).

Equation Transcription:

Text Transcription:

f(y)=

(1 over theta)e^-y/theta, y>0

0, elsewhere.

E(Y)

V(Y)

ANSWER:

Solution 136E

Step1 of 3:

Let us consider a random variable Y it follows exponential distribution with density function:

 

Here our goal is:

a). We need to find the moment-generating function for Y .

b). We need to find E(Y ) and V (Y ).


Step2 of 3:

a).

We know that the probability distribution function of gamma distribution is:

Comparing this probability distribution func

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