Examples10/26

(X,Y) uniform on $\{(x,y):0<x<y<1\}$, what are the joint and marginal densities?

f(x,y)=c, 0<x<y<1

, as the area is $\frac{1}{2}$ take c=2
Marginal Densities

$$f_X(x)=\int_{-\infty}^{\infty}f(x,y)dy=\int_{y=x}^1 2 dy= 2(1-x)$$

$$f_Y(y)=\int_{-\infty}^{\infty}f(x,y)dx=\int_{0}^{x=y} 2 dy= 2y$$

We remarked that :

$$f_X(x)f_Y(y) \neq f(x,y)$$

Here is what this means: