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# Homework Solutions

## Utah State University ECE 6010 Stochastic Processes Homework # 5 Solutions

1. Let and (independent). Let and . Show that and

Here we have,

Therefore,

Also, , therefore

Jacobian is given by

Therefore,

Now,

 and

2. If and are independent and , show that has density .

Here and are independent and , therefore we have,

Now,

As, the above integral exists from 0 to 1 only, therefore

3. Let and . Let . Show that

Such a random variable is said to be chi-squared distributed with two degrees of freedom.

Taking derivatives,

4. If and , show that is exponentially distributed.

We have,

So,

Therefore, we have

5. Let be a monotone increasing function and let . Show that

Suppose that . Then

But when , the intersection of and is . So

It works oppositely when .

6. Let . Show that

Solution: Introduce the auxiliary variable . Then

so

Then integrate out to get .
7. Let and be independent with and . Determine the density of:
1. .

Therefore,

2. .

Therefore,

So,

8. Show that if are i.i.d. , then has a Cauchy distribution,

We have

Using the result from the notes we have with

Then