# All of Statistics - Chapter 7 Solutions

## 1.

Note that

Moreover,

By the bias-variance decomposition, the MSE converges to zero. Equivalently, we can say that converges to in the L2 norm. Since Lp convergence implies convergence in probability, we are done.

*Remark*.
For each , is a random variable.
The above proves only that each random variable converges in probability to the true value of the CDF .
The Glivenko-Cantelli Theorem yields a much stronger result; it states that 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 . The standard error is . We can estimate the standard error by . By the CLT,

and hence an approximate 90% confidence interval is . The second part of this question is handled similarly.

## 3.

TODO

## 4.

By the CLT

Equivalently,

Or, more conveniently,

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

## 5.

Without loss of generality, assume . Then,

## 6.

By the results of the previous question,

We can use the estimator

An approximate confidence interval is .

*Remark*. The closer is to zero or one, the smaller the standard error.

## 7.

TODO

## 8.

TODO

## 9.

This is an application of our findings in Question 2. In particular, we use the estimate . A confidence interval for this estimate is where

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

## 10.

TODO