Let $(X, Y )$ be a random pair with the density $f(x,y) = c(y-x)^2, 0<x<y<1$ and $0, elsewhere$ for some constant $c > 0$.

Question

Let $(X, Y)$ be a random pair with the density $$ f(x,y) = \begin{cases} c(y-x)^2, & 0<x<y<1\\ 0, & \text{elsewhere} \end{cases} $$ for some constant $c > 0$. Find $c$.

So far I'm assuming I integrate like so $\int_{0}^{x}$ $\int_{y}^{1}$ ${c(y-x)^2 dydx}$ but I'm not exactly where where to go from here.

Answer 1

hint

you need to set up the integration correctly. from your bounds, you can take the inner integral varying $y$ from $x$ to $1$, and the outer integral varies $x$ from $0$, can you find to what upper limit? then integration yields

$$ 1 = c \int_0^{???} \int_x^1 (y-x)^2 dy dx $$

and if you integrate it becomes an arithmetic problem.