Joint cdf of guaranteed event

Question

I need someone to sanity check my maths here, I think I'm right but need some help! Let X be a continuous random variable taking values in R+ and let Y be a discrete random variable taking values in {0,1} (to simplify the equation). X and Y are not independent.

Are the following statements true:

i) $F_{X,Y}(x,1) = P(X \leq x, Y \leq 1) = P(X \leq x)$

ii) $f_{X,Y}(x,1) = f_X(x)$

where $f_{X,Y}$ is the joint 'density' defined as in this wiki page.

i) is just the marginal as $Y \leq 1$. I think ii) follows as this is just simple differentiation/subtraction of each case (continuous and discrete)?

Thanks for the help in advance! By the way if independence makes a difference to the answer I'd appreciate if someone could explain why!

EDIT: Simplfied question and key points.

Answer 1

You cannot differentiate with respect to $y$ so there is not a density $f_{X,Y}(x,y)$ in the standard sense.

$f_{X}(x)$ may be meaningful as the derivative of $F_X(x)=\mathbb P(X\le x)=F_{X,Y}(x,1)$. Is this what you want?

Or are you looking for something like $f_{X\mid Y=1}(x)$, which should integrate to $1$,

or like $f_{X\mid Y=1}(x) \mathbb P(Y=1)$, which should integrate to $\mathbb P(Y=1)$?