TODO

TODO

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 $X_1$ once, sampling $X_2$ three times, sampling $X_3$ zero times, and sampling $X_4$ once. In general, there are $n$ stars and $n-1$ bars, and hence the total number of configurations is $(2n - 1)!/(n!(n-1)!)$.

## 5.

First, note that

Therefore, by the tower property, $\mathbb{E}[\overline{X}_{n}^{*}]=\mathbb{E}[X_{1}]$. Next, note that

The above can also be expressed as $S_{n}(n-1)/n^{2}$ where $S_{n}$ is the unbiased sample variance of $(X_{1},\ldots,X_{n})$. Next, note that

where $\mu = \mathbb{E}[X_{1}]$ and $\sigma^{2} = \mathbb{V}(X_{1})$. Now, recall that for any random variable $Y$,

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 $n$.

TODO

## 7.

### a)

The distribution of $\hat{\theta}$ is given in the solution of Question 2 of Chapter 6.

TODO

### b)

Let $\hat{\theta}^*$ be a bootstrap resample. Then,

TODO