Sum of Gamma distributions

Question

The lifetime of a particular brand of batteries is known to have a gamma distribution. Tests on a large sample of these batteries show a mean lifetime of 480 hours and a standard deviation of 96 hours.

Some questions I require help on

1. Show that the moment estimators for the gamma distribution parameters are alpha(hat)=25, and lambda(hat)=5/96. Hence write down and simplify the density function for the lifetime (in hours), using these values.

2. What is the exact distribution of the sum of the lifetimes of 20 of these batteries? Find the exact distribution of the sample mean?

3. Using a normal approximation, find the probability that the sample mean lifetime from these 20 batteries will be less than 460 hours.

4. Based on the parameters of the exact distribution in (b), does the normal approximation in (c) seem reasonable?

I have been able to do #1 ok, but are getting stuck on the rest, if someone could help that would be great.

Answer 1

For 2) think about conditions for which was of gammas equal a gamma and when it does how do the parameters change.

For 3) Once you have the parameters for 2) you can get the mean and variance for this distribution. Those will be the parameter values for teh approximating normal distribution.

Now for 4) you can compare the exact distribution for the sum of the gammas that you got in 2) and compare it to the noraml approximation that you got in 3). To determine if it is reasonable compare some percentiles such as the 90th and 95th.