# All of Statistics - Chapter 3 Solutions

## 1.

Let be the number of dollars at the -th trial. Then,

By the rule of iterated expectations, . By induction, .

## 2.

If , then and hence .

The converse is more complicated. We claim that whenever is a nonnegative random variable, implies that . In this case, it is sufficient to take to conclude that .

To substantiate the claim, suppose . Take . Then,

It follows that for all . By continuity of probability,

## 3.

Since , it follows that . Therefore,

## 4.

Note that where are IID. It follows that and .

## 5.

Let be the number of tosses until a heads is observed. Let denote the result of the first toss. Then,

Solving for yields .

## 6.

## 7.

Integration by parts yields

Define . Note that converges pointwise to as . Moreover, is monotone increasing. The desired result follows by Lebesgueâ€™s monotone convergence theorem.

## 8.

The first two claims follow from

and

As for the final claim, note that

and hence

Next, note that and . Moreover,

and hence . Substituting these findings into the equation above yields , as desired.

## 9.

TODO (Computer Experiment)

## 10.

The MGF of a normal random variable is . Therefore, and

## 11.

### a)

This was already solved in Question 4.

### b)

TODO (Computer Experiment)

## 12.

TODO

## 13.

### a)

Let denote the result of the coin toss. Then,

### b)

Similarly to Part (a),

Therefore, .

## 14.

The result follows from

## 15.

First, note that . Moreover,

and

Therefore, .

## 16.

In the continuous case,

Taking yields . The discrete case is similar. A more general notion of conditional expectation requires Radon-Nikodym derivatives.

## 17.

By the tower property,

and

The desired result follows from summing the two quantities.

## 18.

Since

and by the tower property, .

## 19.

Unlike the distribution of , the distribution of is concentrated around . As increases, so too does the concentration.

## 20.

For a vector with entries ,

For a matrix with entries , define the column vector as the transpose of the -th row of . Then,

Therefore, .

Next, using our findings in Question 14,

As before, we can generalize this to the matrix case by noting that

Therefore, .

## 21.

If , then

and . Therefore,

## 22.

### a)

Note that . Moreover, and . Since , and are dependent.

### b)

If , then and hence . Therefore, trivially. Moreover,

## 23.

Let . The MGF of is

Let . Then,

Therefore, the MGF of is .

Lastly, let . Then,

is finite whenever . Therefore, under the same condition, the MGF of is .

## 24.

Suppose . Then,

and hence

Since this is the MGF of a Gamma distribution, it follows that the sum of IID exponentially distributed random variables are Gamma distributed.