Problem on Statistics - Functions of Random Variables

Question

I have completed part (a) and (b),

(a) I did a double integral from 0 to 1, c=4

(b) I showed the marginal PDFs of X and Y and the product of the two PDF = the joint PDF, thus proving independence

(c) For part C, I have no idea how to start. I've been looking at my notes on functions of random variables but I'm not sure if I'm on the right track.

Any help would be appreciated~ thank you

Answer 1

Hint: You may do the following to get an idea what to do:

• Your distribution lives on $[0,1]\times[0,1]$.
• Draw the square and draw the line $y = 2x$
• Check the region of $y < 2x$ which belongs to that square.

Answer 2

Guide:

Drawing a picture helps.

If you fix $y$, you $x$ values runs from $\frac{y}2$ to $1$.

$y$ takes values from $0$ to $1$. Integrate the pdf over the corresponding region.