Question

A Costco employee performs a total of 60 movements of products within the store during a working
day. The duration of each movement follows an exponential distribution with mean 7 minutes, and the
movements are considered to be independent. What is the probability that the employee will finish his work before eight hours?

Answer 1

Here each movement follows independently exponential distribution with mean 7minutes.

So, 60 movements will follow a gamma distribution with shape parameter 60 and scale parameter 7 minutes.

Now the probability that the employee will finish his work before 8 hours i.e. 480 miniutes is,

P[X < 480] {where X follows Gamma(shape = 60, scale = 7)}

= 0.8644278 [Using R code pgamma(480,60,1/7)]

[NOTE: You may ask why we are taking 1/7 in R code. Actually in R the function pgmma() takes the parameter "rate" instead of "scale" and the relation is rate = 1/scale ]