Part B (i):

We are given that . And given that , we need the distribution of

Step 1: Determine the domain of the random variable . Since is at most , is at most . And similar, because is at least , is at least .

So as well.

Step 2: find the CDF of :

Step 3: find the pdf of

And since and we concluded that , it must be the case that:

Step 4: Vary that indeed this is a pdf:

as desired.

Part B (ii):

This one is a bit tougher. But we can proceed as before:

Step 1: The boundaries of are . So if we let , then the boundaries of become and positive infinity.

So

Step 2: find the CDF of :

Step 3: find the PDF of :

Step 4: verify that this is indeed a valid pdf:

as desired.

Part C:

We proceed as before by finding the cdf of :

Now, here's the trick. Since we have defined or that represents the larger of and , it means that whenever is less than a constant , then it must be the case that both and are less than that constant . So we can determine that:

(by independent of and )

We know that and that

So

We conclude that the cdf of is:

Then, of course, the pdf of is:

Verify that this is a valid pdf and you are done.

Part A:

I am too tired lol...just scroll to the section that says "bivariate case" at: http://en.wikipedia.org/wiki/Bivariate_normal

Plug in the appropriate variables into the formula...