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Suppose $X_1,X_2,\ldots,X_{n}$ be a random sample of $B(20,.5)$. How can find limit distribution $$Y_n=\displaystyle\frac{1}{n}\sum_{i=1}^n (X_i-10)^2$$

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So many questions, so few accepted answers. – Dilip Sarwate Apr 19 '12 at 13:24
up vote 0 down vote accepted

Hint: If you were asked this question, I am sure that you have already done the Central Limit Theorem. To use that, all you need to know is the mean and variance of $(X_i-10)^2$.

The mean of $(X_i- 10)^2$ is undoubtedly already known to you.

For the variance of $(X_i-10)^2$, which you also need, you can use the moment generating function of the Binomial, if you have seen moment generating functions. If you have not, you can let $Y_i=(X_i-10)^2$. We want $E(Y_i^2)-(E(Y_i))^2$. The term $E(Y_i^2)$, if you can think of nothing better, can be found by summing $21$ terms that are not very hard to compute.

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