# All of Statistics - Chapter 8 Solutions

## 1.

TODO

## 2.

TODO

## 3.

TODO

## 4.

This is a stars and bars problem (or, equivalently, an “indistinguishable balls in distinct buckets” problem). For example, the configuration `★|★★★||★`

corresponds to sampling once, sampling three times, sampling zero times, and sampling once. In general, there are stars and bars, and hence the total number of configurations is .

## 5.

First, note that

Therefore, by the tower property, . Next, note that

The above can also be expressed as where is the unbiased sample variance of . Next, note that

where and . Now, recall that for any random variable ,

Therefore, by the tower property,

Applying this to our setting,

As such, we can conclude that

where the asymptotic is in the limit of large .

## 6.

TODO

## 7.

### a)

The distribution of is given in the solution of Question 2 of Chapter 6.

TODO

### b)

Let be a bootstrap resample. Then,

## 8.

TODO