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Thread: Chi-Square stuff?

  1. #1
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    Chi-Square stuff?

    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! (:
    Last edited by panda*; May 26th 2009 at 07:13 AM. Reason: mistake in question (i).
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  2. #2
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    Quote Originally Posted by panda* View Post
    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 $\displaystyle M^2_{1}$. 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 $\displaystyle M_{X^2}(u)=\mathbb{E}(e^{uX^2})$

    (ii) Calculate the moment generating function $\displaystyle M_{S_d}(t)$ of the sum $\displaystyle S_d = X^2_1 + ... + X^2_d$.
    MGF of a sum of independent rv is the product of their mgf.
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  3. #3
    MHF Contributor matheagle's Avatar
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    I'm not sure what you mean by $\displaystyle M_1^2$ in the first part.

    Do you mean $\displaystyle X_1^2$ ?

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

    This may help you...http://en.wikipedia.org/wiki/Noncent...e_distribution
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  4. #4
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    Quote Originally Posted by matheagle View Post
    I'm not sure what you mean by $\displaystyle M_1^2$ in the first part.

    Do you mean $\displaystyle X_1^2$ ?

    These are just noncentral $\displaystyle \chi^2$ 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!
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  5. #5
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    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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