Solve for c of PDF $f(x,y)=cxy1_{0<x<y<1}$

Question

Let (X,Y) be a random vector in R2 with density $f(x,y)=cxy1_{0<x<y<1}$.

1. Find $c$ and $P (X + Y < 1) $

2. Are $X$ and $Y$ independent? Find the density of $(X/Y, Y )$.

3. Are $X/Y$ and $Y$ independent? What is the conditional density of $X$ given $Y = y$?

Find c part, i think it is just taking the double integral of $f(x,y)$ such that the CDF=1. and $P(X+Y<1)$ is just setting $F(x,y)=1$. I hope this is right but for (2) and (3) how do i find out if they are independent or not? do i need to find the density first and then use the property of independence $(P(A,B)=P(A)P(B))$

Answer 1

HINTS

As for $c$, your idea is right.

For $\mathbb{P}[X+Y<1]$, you need to compute $$\iint_A f(x,y)dA$$ with $A = \{0 < x < y < 1 \text{ and } x+y < 1\}.$

If $V$ and $W$ are independent random variables with joint pdf $f_{V,W}$ and individual pdfs $f_V$ and $f_W$ then $$f_{V,W}(v,w) = f_V(v) f_W(w).$$

UPDATE

Since your pdf contains a term $\mathbb{I}_{(0 <x<y<1)}$ which is an unseparable mixture of both $x$ and $y$, then $X$ and $Y$ are not independent.