Solved – Find the variance of p-hat

We have not discussed $hat p$ in my probability and statistics course and a problem involving it is on our hw this week after learning about discrete distributions. The problem states "Let the random variable $Ysim text{Binomial}(n,p)$ and let $hat p = frac{Y}{n}$.

a. Find the mean of $hat p$.

b. Find the variance of $hat p$.

c. Use this and Chebyshevs theorem limit as $n$ goes to infinity of $Pr(verthat p-pvert < a)$ for any $a>0$.

I was able to find the mean of $hat p$ as $p$ and I know that the variance of $hat p$ should be $p(1-p)/n$ but I have been unable to prove that or do part c.

a.

$$ mathbb{E}(hat{p}) = mathbb{E}left(dfrac{Y}{n}right)= dfrac{np}{n}= p$$

b.

$$ operatorname{Var}(hat{p}) = dfrac{operatorname{Var}(Y)}{n^2} = dfrac{p(1-p)}{n} $$

Note here that

$$lim_{nrightarrow infty} operatorname{Var}(hat{p}) =lim_{nrightarrow infty} sigma^2 = 0$$

c. I'll assume you mean Chebyshev's inequality

The inequality says

$$ operatorname{Pr}(vert hat{p} – p| > ksigma) leq dfrac{1}{k^2} $$

In your case $alpha = ksigma$ so $1/k = sigma/alpha$. Since $alpha$ is non -zero, we have no problems thus far. Substituting, we have

$$ operatorname{Pr}(vert hat{p} – p| >a) leq dfrac{sigma^2}{alpha^2} $$

If $alpha$ does not vary with $n$, what can you conclude about

$$lim_{nrightarrow infty} operatorname{Pr}(vert hat{p} – p| >a)$$

Hint: See the law of large numbers

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