Combining Exponential Distributions for Concession Stand Wait Times

[rhyno89]

Homework Statement

A concession stand serves customers with each customer starting as soon as the prior one finishes. The wait times are from an exponential distribution with mean = 2 minutes. Therefore the Total time is the summation of xi for i 1 to 4. Find the mean , variance and distribution of T.

Homework Equations

The Attempt at a Solution

I tried combining the pdfs to get the distribution and thought that I could work it from there. My first try was by comibining the mgf but was fairly certain that it would just make it more complicated so I shifted to trying to work with the exponential to the 4th since it was the same distribution. I got .0625e^-2x.

I thought that this was simple enough but this way they it could not have been another exponential distribution since the value for lambda was different throughout the equation.

As of now I was thinking that it is easier than it looks and the mean of this is simply (mu * n) which would give me 8 minutes.
The variance would then be 1/(.5^2) for each one and since they are independent would be 4*4=16

This seems not only way too easy but also way too high for a variance

Any tips to get me on the right track would be very appreaciated

 

[rhyno89]

what i did was just combine the means since they are independent...resulting in a total mean of 8 minutes. This resulting in a lambda of 1/8 and therefore a variance of 1/64. Finally it would have a pdf of 1/64e^-x/64...

im pretty sure that's right but can anyone clarify?