# 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.

Contents

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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