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 be d independent normally distributed rndom variables with mean 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 (t) of the random variable . All the working must be given.

(ii) Calculate the moment generating function of the sum .

Define a Chi-square distribution with d degrees of freedom as a Gamma distribution with parameters .

Let be a sequence of mutually indepedent standard normal random variables and consider .

(iii) Deduce that has a Chi-square distirbution with d degrees of freedom. Let N be a Poisson random variable with paramater , independent of the . Define

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

(v) Calculate the probability density function of the random variable Z when .

Thanks again! (: