Let X and Y have joint density $f(x,y)=2e^{-x−y}$ over the infinite triangular region where x>0 and 0<y<x.

Let $X$ and $Y$ have joint density $f(x,y)=2e^{-x-y}$ over the infinite triangular region where $x>0$ and $0<y<x$.

a) Calculate $P(X+Y<1)$.

b) Find the marginal density function $f_x(x)$.

c) Find the conditional density $h(y|x)$ when $x=2$.

d) Define $U=Y/X$ and $V=X^2$. Find the joint pdf $g(u,v)$ of $U$ and $V$. (The bounds on $U$ and $V$ are $0<u<1$ and $v>0$).

My attempt

(a) $P(X+Y<1)=\int_{0}^{0.5}\int_{y}^{0.5}2e^{-x-y}dxdy+\int_{0.5}^{1}\int_{1-x}^{0.5}2e^{-x-y}dydx=-\frac{1}{e}+0.60107$

(b) $f_x(x)=\int_{0}^{x}f(x,y)dy=2e^{-x}-2e^{2x}$.

(c)$h(y|x)=\frac{f(x,y)}{f_x(x)}$

$h(y|2)=\frac{f(2,y)}{f_x(2)}=\frac{2e^{-2-y}}{2e^{-2}-2e^{-4}}$

(d) $U=\frac{Y}{X} \Rightarrow Y=XU, X=\sqrt{V}$

$f(\sqrt{v},xu)=2e^{-\sqrt{v}-xu}$

$J(u,v)=\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}=\begin{vmatrix} 0 & \frac{1}{2\sqrt{v}} \\ x & 0 \end{vmatrix} = (0)(0)-(\frac{1}{2\sqrt{v}})(x)=-\frac{x}{2\sqrt{v}}$

$\therefore g(u,v)=f(\sqrt{v},xu)|J|=(2e^{\sqrt{v}-xu})(\frac{x}{2\sqrt{v}})$

Can someone verify thanks.

Solution 1:

a) is wrong

$$\mathbb{P}[X+Y<1]=2\int_0^{0.5}e^{-y}\left[\int_y^{1-y} e^{-x} dx \right]dy=1-\frac{2}{e}$$

The explanation of the correct bounds is evident when doing a drawing

b) correct

c) correct, can be further simplified

d) You cannot have $f(u,v)$ still in function of $x$

The jacobian is 0.5 thus

$$f_{UV}(u,v)=e^{-\sqrt{v}(1+u)}$$

complete it with its joint support