1.

By Lemma 2.15, . Since is right-continuous, .

2.

By Lemma 2.15,

and

3.

1)

Since is monotone, we can write where is some strictly increasing sequence converging to . Let so that . By continuity of probability, .

2)

By additivity, . The desired result follows by moving some terms around.

3)

Taking complements, .

4)

If is continuous, for all by Part 1. The desired result follows from combining this fact with the findings from Part 2.

4.

a)

We can express the CDF using indicator functions:

b)

Since and , it follows that . Next, let . Then,

For ,

5.

Suppose and are independent. Then,

To establish the converse, suppose that . For a subset of the support of and a subset of the support of ,

6.

Note that

7.

Since

it follows that

In particular, when and have the same distribution , .

8.

Let . First, note that and . Moreover, for .

9.

For , . Therefore, .

10.

If and are independent, then

under some lax conditions on and (Borel measurable).

11.

a)

The two variables are dependent because

b)

The two variables are independent because

is decomposable into the form .

12.

If and admit a joint density satisfying , then

The marginal distribution for is where . It follows that . We can similarly define to find that . Moreover, and hence . It follows that , as desired.

13.

a)

Note that

Taking derivatives,

b)

TODO (Computer Experiment)

14.

Let . Then, and hence .

15.

For ,

For all ,

16.

Note that

Moreover,

As per the hint,

Letting , combining these facts yields

17.

First, note that

Therefore,

It follows that

18.

TODO (Computer Experiment)

19.

Let be strictly increasing with differentiable inverse . Let be a continuous random variable. Then, for ,

and hence . If was instead strictly decreasing, then

and hence . Since a strictly decreasing function has a strictly decreasing inverse, it follows that and hence we can summarize both cases by .

20.

Let . Then, and . For , . The region

is either a triangle or a right trapezoid depending on whether or . By covering these case separately, one can derive and , respectively. It follows that

Let . Then, . For , . The region

is either a triangle or a right trapezoid depending on whether or . By covering these cases separately, one can derive and , respectively. It follows that

21.

Since

it follows that .