probability function for poisson random variable A poisson random variable has a mean of x=6.25. A random sample of this variable is drawn. What is the probability function for sum S = $\sum_{i=1}^n X_i$, as specifically as possible.
I have no idea how to attack this. 
 A: Recall that a random variable $W$ is Poisson with parameter $\lambda$ if and only if the mgf of $W$ is equal to $\exp(\lambda(e^t-1))$.
Our $X_i$ therefore all have mgf equal to $\exp((6.25)(e^t-1))$.
The mgf of a sum $X_1+X_2+\cdots+X_n$ of independent random variables is the product of the individual mgf.
Thus in our case the mgf of $X_1+X_2+\cdots+X_n$ is $\left(\exp((6.25)(e^t-1))\right)^n$.
This is equal to $\exp((6.25n)(e^t-1))$. That has shape $\exp(\lambda(e^t-1))$, where $\lambda=6.25n$.  
Thus by the first paragraph of this post, $X_1+X_2+\cdots+X_n$ has Poisson distribution with parameter $6.25n$.
If we want to be really specific, we can write
$$\Pr(S=k)=e^{-6.25n}\frac{(6.25n)^k}{k!}.$$ 
Remark: The moment generating function, and its close relative the characteristic function, can be very useful tools in recognizing that a random variable has a certain distribution. To use these tools, we need to have a familiarity with the mgf (or characteristic functions) of commonly occurring distributions.  
