Limiting Distribution

**problem** · December 30th 2012, 08:55 PM

Let $X_i,i=1,2,3,...$ be independent Bernoulli( $\frac{1}{2}$ ) random variables and let $Y_n=\frac{\sum_{1=1}^nX_i}{n}$ . Find the limitiing distribution of $W_n=n(Y_n(1-Y_n)-\frac{1}{4})$

**Siron** · December 31st 2012, 05:37 AM

First, let's find the distribution of $Y_n$ , this can be done by computing the characteristic function $\phi_{Y_n}(t) = \displaystyle \phi_{\frac{1}{n}\sum_{i=1}^{n}}(t)=\phi_{\sum_{i= 1}^{n }X_i}\left(\frac{t}{n}\right)$ and because $\forall i \in {1,\ldots,n\}$ we have given that $X_i \sim B\left(\frac{1}{2}\right)$ are independent random variables we obtain $\phi_{Y_n}(t) = \prod_{i=1}^{n} \phi_{X_i}\left(\frac{t}{n}\right)$ . 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 $W_n$ , in fact we want to find the distribution function $F_{W_n}(x) = P(W_n \leq x)$ . Can you compute this?

Thread: Limiting Distribution

LinkBack

Thread Tools

Display

Limiting Distribution

Re: Limiting Distribution

Similar Math Help Forum Discussions

Limiting distribution = Initial Distribution MC

Limiting distribution involving chi square distribution

limiting distribution

What are the Limiting Distribution and Limiting Probabilities

Limiting Distribution

Search Tags