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