My Math Forum Let $X$ be the total service time for $10$ customers. Estimate the probability that

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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: 2,369 Thanks: 1273 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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