Hmm, I am looking at this question but it wasn't covered in the lecture so I am looking forward to learn up how to do this kind of questions!

Let $\displaystyle X_1, ..., X_d$ be d independent normally distributed rndom variables with mean $\displaystyle m \in \mathbb R$ and unit variance. The goal of this problem is to study the sum of the squares of these random variables.

(i) Determine the moment generating function $\displaystyle M_{X^2}$(t) of the random variable $\displaystyle X^2_{1}$. All the working must be given.

(ii) Calculate the moment generating function $\displaystyle M_{S_d}(t)$ of the sum $\displaystyle S_d = X^2_1 + ... + X^2_d$.

Define a Chi-square distribution with d degrees of freedom as a Gamma distribution with parameters $\displaystyle \gamma (\frac{1}{2}, \frac{d}{2})$.

Let $\displaystyle Y_1, Y_2 ...$ be a sequence of mutually indepedent standard normal random variables and consider $\displaystyle S'_d = Y^2_1 + ... + Y^2_d$.

(iii) Deduce that $\displaystyle S'_d$ has a Chi-square distirbution with d degrees of freedom. Let N be a Poisson random variable with paramater $\displaystyle \lambda > 0$, independent of the $\displaystyle Y_{j}'s, j = 1,2, ...$. Define

$\displaystyle Z = \sum^{d+2N}_{j=1} Y_{j}^2$

(iv) Determine the moment generating function of Z, and deduce the value of the parameter $\displaystyle \lamda$ such that Z has the same distribution as the sum $\displaystyle S_d$.

(v) Calculate the probability density function of the random variable Z when $\displaystyle \lambda = \frac{m^2d}{2}$.

Thanks again! (: