"Let $X_{1},X_{2},\cdots$ be a sequence of independent random variables such that $X_{n}$ is binomial with parameters $2n-1$ and $p=\frac{1}{2}$. If

$$Y_{n} = \frac{2(X_{1}+X_{2}+\cdots +X_{n})}{n}-n$$

show that the sequence $Y_{n}$ converges in distribution to a standard Normal distribution.

My attempt

I believe I'm supposed to use the Strong Law of Large numbers to prove the following:

$$\lim_{n\rightarrow \infty}\frac{2(X_{1}+X_{2}+\cdots +X_{n})}{n}-n=0$$

I've figured that $E[X_{i}]=n-\frac{1}{2}$, but when I try to apply it, I get something like this:

$$2E[X_{i}]=2(n-\frac{1}{2})=2n-1\\ \Rightarrow \lim_{n\rightarrow \infty}(n-1)=\infty$$

Edit: $E[X_{i}]=2i-\frac{1}{2}$ for every $1\leq i \leq n$, which makes the above expectation moot, but I also don't know how to simplify $Y_{n}$ using this expectation. It might be I can't use Law of Strong Number since it means each expectation increases as $n$ increases.

I'm also not sure of how to prove that the variance equals 1.

  • 1
    $\begingroup$ You know (are supposed to use) the Central limit theorem? $\endgroup$ – leonbloy Nov 9 '13 at 0:20
  • $\begingroup$ I'll probably need it to completely solve this to prove it's normally distributed. $\endgroup$ – IndustrialPioneer Nov 9 '13 at 0:48

Forget about the strong law of large numbers. To solve this problem, you need to put the following ideas together.

  • The sum of independent Binomial$(n_i,p)$ random variables is a Binomial$\left(\sum_i n_i,p\right)$ random variable

  • The mean of a sum of random variables is the sum of the means

  • The variance of a sum of independent random variables is the sum of the variances of the random variables

  • If $Z$ is a Binomial$(n,p)$ random variable and $n$ is large, then the cumulative probability distribution function (CDF) of $\displaystyle \left(\frac{Z - E[Z]}{\sqrt{\operatorname{var}(Z)}}\right)$ is well-approximated by (in fact converges to) $\Phi(\cdot)$, the CDF of the standard normal random variable. This is sometimes referred to as the DeMoivre-LaPlace approximation theorem and is a special case of the Central Limit Theorem that leonbloy suggested to you in a comment.

I assume that you do know, or know how to compute, the mean and variance of a binomial random variable.


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