# Product of two probability distributions and joint PDFs

I have two questions.

1) I am given the probability distributions for X and Y (last picture below), and I have to find the probability distribution for Z = XY. Do I simply multiply the probabilities of X and Y or do I need to find their pdfs first? (Also, this was just the distribution of a fair coin being tossed.

2) For the second question (the picture above), I am given a joint PDF for X and Y and I am asked to find their marginal PDFs. I've done for X, and I'd like to know if I've done it right. If yes, then I can continue with Y myself.

Thanks in advance. (click on pictures to see full questions)

Table for the joint distribution of X and Y

• Are $X$ and $Y$ independent, in part (1)? – Anna SdTC Feb 6 '17 at 21:43
• @AnnaSdTC Nope, they are not independent – NDZS Feb 6 '17 at 21:47
• In that case you need to supply information about their dependence: so far, you have only presented the marginal distributions, but you need the joint distribution. – whuber Feb 6 '17 at 22:52
• @whuber I have added a (picture) table of the joint distribution together with the marginal distribution. Though I'm not too sure if I've done it right – NDZS Feb 6 '17 at 22:59

For the second question, you're slightly off however. Your bounds for the integral should not equal y and 0, but rather 1 and x. Given that you are integrating with respect to y, you should also take the bounds of y. This gives us the following: $$f(x)=\int_x^1 \! 8xy \, \mathrm{d}y = 4x-4x^3$$
For $f(y)$, you can simply repeat the same process. Just be careful when choosing the range of values to integrate over. Hope this helps!