I would really appreciate help on this homework problem.
The random variable X is the waiting time till the occurrence of the first event in a poisson process with expected waiting time beta = .7. Consider a sample of size N = 40 from X.
a) find Prob(.6 <= Xbar <= .8)
b) find the 95 percentile for Xbar
I know that I would start with the gamma distribution(1,.7). After that, I am not sure how to incorporate the sample size into the problem. My class is taught using only excel so I don't really have a textbook to look at or anything. So far we have been mostly dealing with normal distributions and I know for a normal distribution I would divide the standard deviation by the square root of the sample size to get the distribution of the sample mean. Is there anything like that for a gamma distribution? Where I can simply get the distribution of the sample mean.
I do know how I would set these problems up given the distribution. I just need help on getting the correct distribution of the sample mean.
Best Answer
Have you been taught the Central Limit Theorem, which states for $n$ sufficiently large (usually $n\ge30$) that x-bar is normally distributed for any original distribution, and has the same mean, but the S.E.S.M. is calculated using $\frac{\sigma}{\sqrt{n}}$. Remember also, for a gamma distribution with parameter $a,b$ that $\mu=a b,\sigma=\sqrt{a} b$