Use moment generating functions to find the distribution of $Y$?

Question

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 $Y$.

Answer 1

Use the fact that if $M_X(t)$ and $M_Y(t)$ are the MGFs of independent random variables $X$ and $Y$, then the MGF of their sum $X+Y$ is given by $$M_{X+Y}(t) = \operatorname{E}[e^{tX + tY}] \overset{\text{ind}}{=} \operatorname{E}[e^{tX}]\operatorname{E}[e^{tY}] = M_X(t) M_Y(t);$$ that is to say, the MGF of a sum of independent random variables equals the product of the MGFs of each variable. From this, you can deduce that the sum $Y$ of $n$ IID exponential random variables $X_1, \ldots, X_n$ with common mean parameter $\beta$ has MGF $$M_Y(t) = (1 - \beta t)^{-n}.$$ What type of random variable corresponds to such an MGF?

Answer 2

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?