# Thread: Approximating the distribution of a sum of skewed random variables

1. ## Approximating the distribution of a sum of skewed random variables

Work schedules: Suppose the length of time for a particular service (say 7500 mile auto maintenance) has a mean of 1.5 hours and SD equal to 1 hour. Because there are occasional complications, this distribution is skewed to the right, i.e., has a longer right tail than left tail (some services run over 3 hours, but none run under 0 hours). Suppose there are 60 services scheduled for a day (8 hours). There are 12 workers available, each on an eight hour shift. Find the probability that overtime will be required. (Do you think it is OK to use a normal distribution? Why?)

2. Originally Posted by ShaunW
Work schedules: Suppose the length of time for a particular service (say 7500 mile auto maintenance) has a mean of 1.5 hours and SD equal to 1 hour. Because there are occasional complications, this distribution is skewed to the right, i.e., has a longer right tail than left tail (some services run over 3 hours, but none run under 0 hours). Suppose there are 60 services scheduled for a day (8 hours). There are 12 workers available, each on an eight hour shift. Find the probability that overtime will be required. (Do you think it is OK to use a normal distribution? Why?)
Without further information about the distribution of service times you should use Chebeyschev's inequality to investigate this.

(One would sometimes assume that 60 is a large number and use a normal approximation, but this is making an implicit assumption about the distribution which may not be valid but often is adequate).

The total time T for services will have a mean of 90 hours with a SD of ~7.75 hours, the available hours for servicing without overtime is 96 hours.

So the probability that overtime will be needed is: Pr(T>96)

CB