# All of Statistics - Chapter 4 Solutions

## 1.

Chebyshev’s inequality gives . An exact calculation yields instead . To see this, note that and so that

## 2.

## 3.

First, note that . Chebyshev’s inequality yields

Next, note that

Let so that . Then, and . Hoeffding’s inequality yields

Similarly, . It follows that

is tighter than the Chebyshev bound for sufficiently large .

## 4.

### a)

Applying our findings from Question 3,

### b)

TODO (Computer Experiment)

### c)

The length of the interval is . This length is at most if and only if .

TODO (Plot)

## 5.

As per the hint,

## 6.

TODO (Plot)

## 7.

A linear combination of IID normal random variables is itself a normal random variable. Therefore, is a random variable with zero mean and variance . Letting , Mill’s inequality yields

The above is tighter than the Chebyshev bound for sufficiently large .