
Probability and Statistics Basic Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 3rd, 2017, 01:37 PM  #1 
Member Joined: Apr 2017 From: PA Posts: 43 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: CA Posts: 1,297 Thanks: 664 
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 $f_X(t) = \dfrac {1}{20} e^{\frac {t}{20}}$ $\begin{align*} &P[X > 22] \\ \\ &= \displaystyle \int_{22}^{\infty}~f_X(t)~dt \\ \\ &= 1  \int_{0}^{22}~f_X(t)~dt \\ \\ &= 1  \int_{0}^{22}~\dfrac {1}{20} e^{\frac {t}{20}} ~dt \\ \\ &= e^{\frac{22}{20}} \\ \\ &\approx 0.333 \end{align*}$ 

Tags 
$10$, $x$, customers, estimate, probability, service, time, total 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Total Probability  marks2014  Probability and Statistics  3  April 30th, 2014 12:23 PM 
probability & time series  sthoriginal  Advanced Statistics  0  February 25th, 2014 07:52 AM 
Total Possibilities of a Probability Tree (with node reuse)  Pherion  Probability and Statistics  1  June 13th, 2013 12:01 AM 
3 probability ( all need total arrangement possible )  rnck  Probability and Statistics  7  November 5th, 2011 08:33 PM 