# Find the distribution of $\overline{Y}$.

Let $\overline{Y}=\frac{\sum_{k=1}^n Y_n}{n}$.

Knowing that $Y_1, Y_2, ..., Y_n$ are independent identically distributed random variables with gamma distribution $\beta=4,\alpha=1.5$, I already found the distribution of $\sum_{k=1}^n Y_n$, which is gamma distribution with $\beta=4,\alpha=1.5n$.

How can I use that knowledge to find the distribution of $\overline{Y}$.

• Since you have the density of $X= \sum_{k=1}^n Y_n$, then define the random variable $Z = X/n$ and apply the change-of-variable formula to obtain its density. – Alecos Papadopoulos Oct 3 '13 at 22:56
• Okay I believe I used the transformation correctly to get that $Z$ has a gamma distribution with β=4/n and α=1.5n. Can someone verify? – Chuck Franklin Oct 4 '13 at 2:04