Question on MGFs & Finding The Distribution of a Sum Suppose that $X_1$, $X_2$, ..., $X_n$ are independent, where each $X_i$ has probability (mass) function $p_i$($x_i$) given as follows:
$p_i$($x_i$) = $\frac{e^{-\lambda}\lambda_i^{x_i}}{x_i!}$ (the parameter $\lambda_i$ differs in the distribution  for each $X_i$ for $x_i$ = 0, 1, ...
What is the distribution of their sum $\Sigma_{i = 1}^{n}X_i$? Prove it using a moment generating function.
Also what is the approximate distribution of $\sum_{i=1}^{5}\frac{(X_i - \lambda_i)}{\lambda_i}$ if $X_1$, $X_2$, ... $X_5$ are very nearly normal?
Can somebody help me out with these two questions? I'm aware of the mgf quality $M_{\Sigma_{i = 1}^nX_i}(t) = M_{x_1}(t) * M_{x_2}(t) * ... * M_{x_n}(t)$ but I'm not sure how exactly  I can use the resulting MGF to get the CDF or PDF. 
 A: Hints: (1) A random variable $W$ has Poisson distribution with parameter $\lambda$ if and only if the distribution of $W$ has mgf $\exp\left(\lambda(e^t-1)\right)$. 
(2) Since $X_i$ has mgf $\exp\left(\lambda_i(e^t-1)\right)$, by independence $\sum X_i$ has mgf the product of the individual mgf. This product simplifies to  $\exp\left((\lambda_1+\lambda_2+\cdots +\lambda_n)(e^t-1)\right)$. 
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(3) Now look again at (1). The result we got through the mgf is that the sum of independent random variables with Poisson distribution has Poisson distribution.
For the second question, from the mgf, or otherwise, we find that the mean of $X_i$ is $\lambda_i$.  Also from the mgf, we find that the expectation of $X_i^2$ is $\lambda+\lambda^2$, and therefore the variance of $X_i$ is $\lambda_i$.
Thus $\frac{X_i-\lambda_i}{\lambda_i}$ has mean $0$ and variance $\frac{1}{\lambda_i^2}\lambda_i=\frac{1}{\lambda_i}$. 
The sum asked about is the sum of $5$ independent nearly normals. So the sum is nearly normal. From the known mean and variance of $\frac{X_i-\lambda_i}{\lambda_i}$ we conclude that  $\sum_{i=1}^5 \frac{X_i-\lambda_i}{\lambda_i}$ is nearly normal, mean $0$, variance $\sum_{i=1}^5 \frac{1}{\lambda_i}$. 
