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## Utah State University ECE 6010 Stochastic Processes Homework # 2 Solutions

1. Suppose is a r.v. with c.d.f. . Prove the following:
1. is nondecreasing.

Let .

So for , which means is nondecreasing.

2. .

3. .

4. is right continuous.

Let for . Note that this is a nested sequence, . We have

by continuity of probability. But , so

Since the limit from the right is equal to the limiting value, we have right continuity.

5. if .

.

6. .

Also, find expressions for , and in terms of .

2. Show that the following are valid p.m.f.s:
1. Binomial: if .

Need to show that . Use the binomial theorem:

with and . Then

2. Poisson: for .

Need to show that .

3. Find the mean and variance of when is
1. ;
Let , therefore

Function has odd symmetr. Integrating an odd function on a symmetric interval -a,a gives zero. Thus,

For variance,

With and , we have and .

Now . So, .

2. Binomial ;
Binomial p.m.f is given by

Therefore,

The first term in the above summation will be zero so we could start it from 1. Also cancelling the common factors of in numerator and denominator.

Making change of variable above we get,

The terms in the summation are just the binomial funciton for trials, and we are summing it over all values of so sum is 1.

Now,

Therefore,

3. Poisson ;
Poisson :

So,

4. Exponential ;
Exponential : .

So,

4. Suuppose that and are jointly continuous. Show that

Therefore,

Now,

5. Suppose that and are jointly Gaussian with parameters . Show that .
In this case we have,

(Hint : Do substitution of variables and Complete the squares)
6. Suppose , and define . Are and uncorreleated? Are and independent? Find the pdf of . Are and jointly continuous?

Note that and . Then

But for a Gaussian with mean zero, all odd moments are 0, so . So , and and are uncorrelated. As , cannot be independent of -- they are functionally related.

Now, X is a continuous r.v. so , then for

where, is the standard unit step function.

X and Y are jointly continuous: Look at the joint CDF:

This is a continuous function of and (as can be realized with a little thought).