Chi-Square stuff?

panda* · May 25th 2009, 06:16 AM

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 $X_1, ..., X_d$ be d independent normally distributed rndom variables with mean $m \in \mathbb R$ 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 $M_{X^2}$ (t) of the random variable $X^2_{1}$ . All the working must be given.

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

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

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

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

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

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

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

Thanks again! (:

**Moo** · May 25th 2009, 11:47 AM

Originally Posted by panda*

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 $X_1, ..., X_d$ be d independent normally distributed rndom variables with mean $m \in \mathbb R$ 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 $M_{X^2}$ (t) of the random variable $M^2_{1}$ . All the working must be given.

See here : http://www.mathhelpforum.com/math-he...513-post2.html, by letting t=0.
Because you're looking for $M_{X^2}(u)=\mathbb{E}(e^{uX^2})$

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

MGF of a sum of independent rv is the product of their mgf.

**matheagle** · May 25th 2009, 01:56 PM

I'm not sure what you mean by $M_1^2$ in the first part.

Do you mean $X_1^2$ ?

These are just noncentral $\chi^2$ random variables, since m is not necessarily zero.

This may help you...http://en.wikipedia.org/wiki/Noncent...e_distribution

panda* · May 26th 2009, 07:14 AM

Originally Posted by matheagle

I'm not sure what you mean by $M_1^2$ in the first part.

Do you mean $X_1^2$ ?

These are just noncentral $\chi^2$ random variables, since m is not necessarily zero.

This may help you...Noncentral chi-square distribution - Wikipedia, the free encyclopedia

Sorry, my bad. I have corrected that mistake!

panda* · May 27th 2009, 03:54 AM

I read through the references notes that you guys have provided but unfortunately I still have no idea where to start on! A kick start first step perhaps? (:

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