Let $\displaystyle X_i,i=1,2,3,...$ be independent Bernoulli($\displaystyle \frac{1}{2}$) random variables and let $\displaystyle Y_n=\frac{\sum_{1=1}^nX_i}{n}$. Find the limitiing distribution of $\displaystyle W_n=n(Y_n(1-Y_n)-\frac{1}{4})$
Let $\displaystyle X_i,i=1,2,3,...$ be independent Bernoulli($\displaystyle \frac{1}{2}$) random variables and let $\displaystyle Y_n=\frac{\sum_{1=1}^nX_i}{n}$. Find the limitiing distribution of $\displaystyle W_n=n(Y_n(1-Y_n)-\frac{1}{4})$
First, let's find the distribution of $\displaystyle Y_n$, this can be done by computing the characteristic function $\displaystyle \phi_{Y_n}(t) = \displaystyle \phi_{\frac{1}{n}\sum_{i=1}^{n}}(t)=\phi_{\sum_{i= 1}^{n }X_i}\left(\frac{t}{n}\right)$ and because $\displaystyle \forall i \in {1,\ldots,n\}$ we have given that $\displaystyle X_i \sim B\left(\frac{1}{2}\right)$ are independent random variables we obtain $\displaystyle \phi_{Y_n}(t) = \prod_{i=1}^{n} \phi_{X_i}\left(\frac{t}{n}\right)$. Now, the only thing you have to do is calculating the characteristic function of a random variable with a Bernoulli distribution. Can you do that?
Second, we want the distribution of $\displaystyle W_n$, in fact we want to find the distribution function $\displaystyle F_{W_n}(x) = P(W_n \leq x)$. Can you compute this?