The question is from Probability and Statistics:
Q. A point is selected at random inside the triangle $T=\{(x,y) : 0\leq y\leq x\leq 1\}$. Assume the point is equally likely to fall anywhere in the triangle. Find the joint and marginal pdf of X and Y.
I tried solving this but was not able to come up to the answer. Please someone help me come to the right answer. Thanks! Have a nice day!
Do a drawing of your problem and realize that $(X,Y)$ domain is the lower triangle of the unit square (the triangle under the line $X=Y$)
As per the fact that the point is equally likely to fall in any poin of the triangle, the bivariate density is uniform, thus its density is the reciprocal of Triangle area
$$f_{XY}(x,y)=2\cdot\mathbb{1}_{0\leq y\leq x\leq1}$$
Now to find the marginals is enough to integrate the other rv
$$f_X(x)=\int_0^x 2dy=2x\cdot\mathbb{1}_{[0;1]}(x)$$
$$f_Y(y)=\int_y^1 2dx=2(1-y)\cdot\mathbb{1}_{[0;1]}(y)$$
To understand well the integration extremes just have focus on the given domain
$$\underbrace{0<y<x}_{\text{y-integration interval}}<1$$
$$0<\underbrace{y<x<1}_{\text{x-integration interval}}$$