Find the joint CDF of X and Y...

Question

Let X and Y be two continuous random variables with joint PDF $f(x,y)=c.e^{-x}$

For $x >0$ and $|y|<x$

Find the value of c and find the joint CDF of X and Y.

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This is my drawing

After integrating for a bit I found c to be $1/2$

So, $f(x,y)=(1/2)e^{-x}$

Since I have 3 pieces (piece no $5,4,3$) with $F(x,y)=0$

I also know that $\int_{0}^{y}\int_{0}^{x}f_{X,Y}(s,t)dsdt$

Working with the first piece I have: $\int_{0}^{y}\int_{0}^{x}(1/2)e^{-x} dxdy$

But working with those bounds doesn't yield the correct answer.

Any tips or help will be greatly appreciated.

At this point, I'm lost as nothing I do look like the answers.

Even the marginal $f_Y(y)$

Answer 1

You just have to write that :

$$ \int_{\mathbb R^2} f(x,y) dxdy = 1 $$

Indeed :

\begin{align*} \int_{\mathbb R^2} ce^{-x} \mathbb 1_{|y| < x}\mathbb{1}_{x>0} dxdy = 1 = \int_{\mathbb R_+} c 2x e^{-x} dx= 1 = 2 c \Gamma(2) = 2c = 1 \end{align*}

Hence :

$$c = \frac{1}{2} $$

Now you can write :

$$ f(x,y) = \frac{1}{2} e^{-x}\mathbb 1_{|y| < x}\mathbb{1}_{x>0} dxdy $$

Thus :

$$ F(t,v) = \mathbb P(X \leqslant t, Y \leqslant v) = \int_{-\infty}^t \int_{-\infty}^v \frac{1}{2} e^{-x}\mathbb 1_{|y| < x}\mathbb{1}_{x>0} dydx$$

Hence :

\begin{align*} F(t,v) &= \int_{-\infty}^t xe^{-x} \mathbb{1}_{\{x >0\}}dx = \int_0^t xe^{-x} dx \end{align*}