# probability – question about the plan of the region when we transform pairs of random variables

Leave $$X$$ Y $$Y$$ They will be two independent random variables.

Leave $$Z = g (X, Y)$$ A function of two random variables.
To find the CDF of $$Z$$ we have

begin {align} P {Z leq z } = & P {g (X, Y) leq z } \ = & iint_ {g (x, y) leq z} ^ {} f_X (x) f_Y (y) dxdy. end {align}

In the books, Inequality. $$g (x, y) ≤ z$$ defines a region in the $$(x, y)$$ airplane.
How do we find this region? For example $$Z = X + Y$$.

Another question is: Why do we move in capital letters? $$(g (X, Y))$$ a lowercase $$(g (x, y))$$?

Also for example the probability that a random variable $$X$$ It is less then or equal to another random variable. $$Y$$ one is
begin {align} P {X leq Y } = & int_ {y = – infty} ^ {+ infty} int_ {x = – infty} ^ {y} f_X (x) f_Y (y) dxdy. end {align}
Because it is not
begin {align} P {X leq Y } = & int_ {y = – infty} ^ {+ infty} int_ {x = – infty} ^ {Y} f_X (x) f_Y (y) dxdy. end {align}
I mean
$$int_ {x = – infty} ^ {Y}$$
do not

$$int_ {x = – infty} ^ {y}.$$