Calculation of joint probability distribution function

Question

Question: (Two Functions of two RVs, Z = g(X,Y), W = h(X,Y)) Let X and Y be two exponentially distributed independent Random Variables with the same parameter (λ). Let Z = X + DY = g(X, Y), and W = DX - Y = h(X,Y) be the two transformed RVs obtained  through linear combination of X and Y RVs respectively, where  where D is a constant. Answer the following questions:

1. The joint PDF of the transformed RVs, Z and W with their proper ranges along with its

proper sketch  

2. Their marginal and and conditional PDFs along with their sketches

3. Are Z and W independent RVs? Briefly explain

4. Are Z and W uncorrelated RVs? Briefly explain

5. If answer to part 4) is no, then find their correlation coefficient, ρ

6. How do the mean and the variance of the RVs Z and W vary with D?

7. Compute their conditional mean and conditional variance

8. Compute their Joint MGF and Joint CF in terms of D

Answer 1

not sure if you're still looking for help with this, but if you are, I'd say that since you have several questions and they're fairly involved, a tutoring session would be the best way to go—and I'd be happy to help if you'd like!

That said, 1 is asking you to find the Probability Distribution Functions for the random variables Z and W. The values of Z are given by the function g(X, Y) and W by h(X, Y). The probability density function of X for an exponentially distributed random variable is f(x) = me^(-mx) where m is the decay parameter, I believe given by lambda in your problem. The sketch is just a rough drawing of the variable's distribution.

The Conditional Distribution Function for exponential rvs is for P(X =< x) = 1 - e^(-mx).

Hopefully that's enough to at least help get you started!