I was attempting to solve this joint probability density function question and wanted to find out if my answers are correct/ would need some more help. The question is:
We are given such a joint probability density function X and Y:
1.Are X and Y independent, moreover are they identically distributed?
2.What is P(X > 0.75 | Y > 0.5)?
3.What is P(Y>X), moreover what is P(Y>=X)?
4.What is E[X]?
The way I solved some of these questions were as follows:
I found the marginal pdf's of X and Y as 12x(1-x)^2 and 12y(1-y)^2 for the boundaries of 0<=x<=1 and 0<=y<=1, respectively. Now, since multiplying the pdfs does not result in the function defined above, it is not independent. Regarding it being identically distributed, I am not sure if it is or not and would love some help.
Since it is defined that 0<=x+y<=1, P(X>0.75 | Y > 0.5) would just result in 0
I solved the probability for P(Y>=X) and got
, is this correct? Moreover, I think that P(Y>X) can be solved by equating it to P(Y>=X)-P(Y=X) but what would the answer for that be?I am completely lost on this one, not going to lie.
Would love some help with the answers! Thanks :D