## 1.

Note that

Moreover,

By the bias-variance decomposition, the MSE converges to zero. Equivalently, we can say that $\hat{F}_n(x)$ converges to $F(x)$ in the L2 norm. Since Lp convergence implies convergence in probability, we are done.

Remark. For each $x$, $\hat{F}_n(x)$ is a random variable. The above proves only that each random variable $\hat{F}_n(x)$ converges in probability to the true value of the CDF $F(x)$. The Glivenko-Cantelli Theorem yields a much stronger result; it states that $\Vert \hat{F}_n - F \Vert_\infty$ converges almost surely (and hence in probability) to zero.

## 2.

Assumption. The Bernoulli random variables in the statement of the question are pairwise independent.

The plug-in estimator is $\hat{p} = \overline{X}_n$. The standard error is $\operatorname{se}(\hat{p})^2 = \mathbb{V}(X_1) / n = p (1 - p) / n$. We can estimate the standard error by $\hat{\operatorname{se}}(\hat{p})^2 = \hat{p}(1 - \hat{p}) / n$. By the CLT,

and hence an approximate 90% confidence interval is $\hat{p} \pm 1.64 \cdot \hat{\operatorname{se}}(\hat{p})$. The second part of this question is handled similarly.

TODO

## 4.

By the CLT

Equivalently,

Or, more conveniently,

Remark. The closer (respectively, further) $F(x)$ is to 0.5, the more (respectively, less) variance there is in the empirical distribution evaluated at $x$.

## 5.

Without loss of generality, assume $% $. Then,

## 6.

By the results of the previous question,

We can use the estimator

An approximate $1 - \alpha$ confidence interval is $\hat{\theta} \pm z_{\alpha / 2} \cdot \hat{\operatorname{se}}(\hat{\theta})$.

Remark. The closer $F(b) - F(a)$ is to zero or one, the smaller the standard error.

TODO

TODO

## 9.

This is an application of our findings in Question 2. In particular, we use the estimate $(90 - 85) / 100 = 0.05$. A $1 - \alpha$ confidence interval for this estimate is $0.05 \pm z_{\alpha / 2} \cdot \hat{\operatorname{se}}$ where

The z-scores corresponding to 80% and 95% intervals are approximately 1.28 and 1.96.

TODO