Random variable conv. in prob. to c. How to find c?

[SithsNGiggles]

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

 

[Ray Vickson]

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

What is $E(W_n)$?

 

[SithsNGiggles]

Judging by $W_n$'s distribution, that would be $n$. Does that mean $c=n$, and if so, is that always the case (that the mean is the constant I'm supposed to find)?

 

[haruspex]

Judging by $W_n$'s distribution, that would be $n$. Does that mean $c=n$, and if so, is that always the case (that the mean is the constant I'm supposed to find)?

No, E(W_n) is not n.

 

[SithsNGiggles]

No, E(W_n) is not n.

Oh right, I forgot that $W_n=\frac{1}{n}\sum\cdots$. Thanks! I meant to say $E(W_n)=1$.

 

[haruspex]

Oh right, I forgot that $W_n=\frac{1}{n}\sum\cdots$. Thanks! I meant to say $E(W_n)=1$.

Right - so all done here?