Joint Cumulative Distribution Function for Lebsegue Measure

Question

Consider the probability triple $([0,1]^2, \mathcal{B}[0,1], Leb_2)$.

Let $X: [0,1]^2 \rightarrow \mathbb{R}$ be random variables defined by

$X(\omega_1, \omega_2) = \omega_1 + \omega_2$,

$Y(\omega_1, \omega_2) = \omega_2 - \omega_1$.

Let $F_{X,Y}$ be the joint cumulative distribution function. Find $F_{X,Y}(0.58,0.35)$.

I have attempted to solve the problem by:

$F_{X,Y}(0.58,0.35)=\mathbb{P}(X \le 0.58, Y \le 0.35)$

$=\mathbb{P}( \omega_1 + \omega_2 \le 0.58, \omega_2 - \omega_1 \le 0.35)$

$=\mathbb{P}( \omega_1 \le 0.115, \omega_2 \le 0.4625)$

$=Leb_2([0,0.115]X[0,0.4625])$

$=0.0535$

However I do not think this is correct.

Answer 1

I am unsure of how I would go about finding the joint cumulative distribution function for X and Y.

If you are unsure this means that you have some ideas to develop... I suggest you to apply a change of variable, calculate the jacobian and find

$$f_{XY}(x,y)=0.5$$

in the following bivariate support

Thus the Joint CDF can be easily derived by integration od the pdf