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August 3rd, 2017, 02:37 PM   #1
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Let $X$ be the total service time for $10$ customers. Estimate the probability that

Assume that the service time for a customer at a bank is exponntially distributed with mean service time $2$ minutes. Let $X$ be the total service time for $10$ customers. Estimate the probability that $X > 22$ minutes.

Attempt: I tried to set up and take the integral $$\int_{22}^{+infinity} λe^{10x}\,\mathrm dx$$ then I eventually did the integral and did an inegtration by parts many times. unfortunately I was unable to figure this out
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August 3rd, 2017, 05:54 PM   #2
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1 customer has average service time 2 min implies 10 customers have average serving time 20 minutes.

Note that the serving time is the inverse of the rate.

The distribution of $X$ is thus

$f_X(t) = \dfrac {1}{20} e^{-\frac {t}{20}}$

$\begin{align*}
&P[X > 22] \\ \\

&= \displaystyle \int_{22}^{\infty}~f_X(t)~dt \\ \\

&= 1 - \int_{0}^{22}~f_X(t)~dt \\ \\

&= 1 - \int_{0}^{22}~\dfrac {1}{20} e^{-\frac {t}{20}} ~dt \\ \\

&= e^{-\frac{22}{20}} \\ \\

&\approx 0.333

\end{align*}$
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