- #1

SithsNGiggles

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## Homework Statement

Let ##Y_1,...Y_n## be independent standard normal random variables.

What is the distribution of ##\displaystyle\sum_{i=1}^n{Y_i}^2## ?

Let ##W_n=\displaystyle\frac{1}{n}\sum_{i=1}^n {Y_i}^2##. Does ##W_n\xrightarrow{p}c## for some constant ##c##? If so, what is the value of ##c##?

## Homework Equations

## The Attempt at a Solution

In response to the first question, I say that ##W_n## has a Chi Square distribution with ##n## degrees of freedom.

For the second, I use a theorem stated in my book that says if ##\text{Var}(\hat{\theta}_n)\to 0##, then ##\hat{\theta}_n\xrightarrow{p}\theta##.

I have that ##\text{Var}(W_n)=\dfrac{1}{n^2}\text{Var}\left(\displaystyle\sum_{i=1}^n {Y_i}^2\right)=\dfrac{2n}{n^2}=\dfrac{2}{n}##, which approaches ##0## as ##n\to\infty##. Thus ##W_n## indeed converges in probability to ##c##.

The problem I'm having now is with finding the value of ##c##. There don't appear to be any examples in my text that describe the process to finding the limiting probability (if that's the right term). Any help is appreciated