I have two questions:

1. Let the joint PDF of X and Y be f(x,y)=2x for 0<x<1 and 0<y<1, 0 otherwise.

I want to find P(X+Y<1 | X<1/2).

First I find the individual distributions f_X(x)=2x and f_Y(y)=1. So P(X<1/2) is 1/4. Then I am stuck. I am not comfortable with the formula for conditional probability with multiple distributions. I am also interested if there is an intuitive answer because y does not appear in the joint PDF.

2. the PDF of X is f(x)=kx if 0<x<(2/k)^(1/2). If the mode of X is (2^(1/2))/4 then the median is what?

So I want to figure out k by using the mode. For continuous distributions I feel like I should take the derivate of the pdf and set it equal to 0 to find the mode, but this does not work, as f'(x)=k, hence is never 0. Instead my idea is that f(x) is maximized at (2/k)^(1/2) as k clearly must be positive. But I am not certain.