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Math Help - Limiting Distribution

  1. #1
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    Limiting Distribution

    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})
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Limiting Distribution

    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?
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