A doorman at a hotel is trying to get three taxicabs for

Chapter 5, Problem 19E

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QUESTION:

A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?

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QUESTION:

A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?

ANSWER:

Step 1 of 4

Given that \(X_{1}, X_{2}, X_{3}\) the arrival of each time.

Let \(Y=X_{1}+X_{2}+X_{3}\)

The formula for MGF is,

\(M_{x}=\frac{1}{1-\theta t}\)

And for Y,

\(\begin{aligned}
M_{Y}(t) & =E\left(e^{t Y}\right) \\
& =E\left(e^{t X_{1}} e^{t X_{2}} e^{t X_{3}}\right) \\
& =\frac{1}{(1-2 t)^{3}}
\end{aligned}\)

The moment generating function of gamma function distribution is \(M(t)=\frac{1}{(1-\theta t)^{\alpha}}, t<\frac{1}{\theta}\).

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