So I'm having trouble grasping bivariate distribution functions. I seem to be struggling with how to determine the upper and lower bounds of my integrals.
So for example, this is a question from my textbook.
(I'm really sorry, I'm not very familiar with MathJax and its syntax so this may be a bit messy)
Let Y1 and Y2 have the joint probability density function given by
f(y1,y2)= {k(1−y2), 0≤y1≤y2≤1,
{0, elsewhere.
a Find the value of k that makes this a probability density function.
b Find P(Y1≤3/4,Y2≥1/2).
I know how to find a double integral, but where I struggle is figuring out what I should be setting my bounds to, and how I should be solving them. I have both a Chegg Study and Slader account, but neither of them really explain their reasoning for selecting their bounds. I feel like I'm on the verge of everything clicking, but I just can't quite get it. Any help would be greatly appreciated.
A probability density function integrates to $1$. The bounds $0\le y_1 \le y_2 \le 1$ mean
$$y_1\le y_2\le 1\\ 0\le y_1\le 1$$
so solve the following for $k$
$$\int_0^1\int_{y_1}^1 k(1-y_2)dy_2dy_1=1$$
The part 2 is given by the integral $$\int_{1/2}^1\int_0^{1/2}f(y_1, y_2)dy_1dy_2+\int_{1/2}^{3/4}\int_{y_1}^1 f(y_1, y_2) dy_2dy_1$$
It would be most helpful if you drew a coordinate system of $y_2$ vs $y_1$ and then the identity line $y_1=y_2$. The support will be the area in the unit square above the identity line (a triangle).