Expected value of mean of N Chi squared random variables

Question

I understand that here they have used law of large numbers. But according to the law the result should equal the expected value of the random variable. But how is the expected value of the r.v only equal to the expectation of the first chi squared r.v. shouldn't it include all the r.v. till n. X1,X2,...,Xn are iid and normally distributed. Then as n approaches infinity what value does summation of Xi squared/n (where i=1,2,..,n) converge to

Answer 1

We are given random variables are iid.Let $Y=X^2$

we know $Y \sim \chi_{1}^2$

$E(Y)=1$

$E\left(\frac{Y_1+Y_2.....+Y_n}{n}\right)=\frac{1}{n}(E(Y_1)+E(Y_2)..+E(Y_{n})))=\frac{n}{n}=1$

Also if you don't know about chisquare result you can use $E(X^2)=V(X)+E(X)^2=1+0$

There you can even use $\frac{1}{n}E(X_{1}^2+X_{2}^2...X_{n}^2)=\frac{n}{n}$

So as $n$ goes large we get $1$