Suppose that $X_1,X_2,\dotsc,X_n$ is a random sample of size $n = 5$ of independent random variables from an exponential distribution with parameter $\beta= 4$.
Suppose $Y= X_1 + X_2 + X_3 + X_4 + X_5$. Use moment generating functions to find the
distribution of $Y$.
I'm just not sure where to start. I think the moment generating function for an exponential distribution is $\frac{1}{1-\beta t}$, which would mean each $X_n$ would have a moment generating function of $\frac{1}{1-4t}$. I'm not sure how I would use this information to find the distribution of $Y$.
Best Answer
The moment generating function of the sum of independent identically distributed random variables is given by the product of moment generating functions: $$m_Y(t)= \prod_{i=1}^5 m_{X_i}=(1-4t)^{-5}$$What distribution has this moment generating function?