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August 3rd, 2017, 01:37 PM  #1 
Member Joined: Apr 2017 From: PA Posts: 45 Thanks: 0  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 
August 3rd, 2017, 04:54 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,429 Thanks: 1314 
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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$10$, $x$, customers, estimate, probability, service, time, total 
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