Joint Density Function Boundaries

conditional probabilitydensity functionprobability

I've been having trouble understanding the following exercise:

The joint density function of the random variables X and Y is:

$$
f(x, y) =\left\{
\begin{array}{ll}
6x,&0<x<1, &0<y<1-x \\
0 &elsewhere \\
\end{array}
\right.
$$

a) Show that X and Y are not independent

b) Find P(X > 0.3|Y = 0.5)

I know that the answers are:

a)

enter image description here

and for Part b:

enter image description here

But what I don't get is how to get the integration boundaries. For example in Part b, I don't see why it should be integrated from 0.3 to 1 – .5, instead of integrating from 0.3 to 1.

How do I know when the boundaries of the integral are different?

Best Answer

Your domain on which $(X,Y)$ are defined is a triangle in $\mathbb{R}^2$ with vertices $(0,0),(1,0)$ and $(0,1)$. If you want to integrate $dx$ first, the boundaries would be $0 < y < 1$ and $0 < x< 1-y$.

This explains the choice of the upper bound of the integral in both parts of the problem.

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