Suppose that $X$ follows a standard normal distribution (such that $X\sim N(0, 1)$). Moreover, suppose that $Y$ is a discrete random variable such that $\mathbb P(Y=-1)=\mathbb P(Y=1)=\frac{1}{2}$. From here, how is the joint probability distribution, $f(x, y)$, or the C.D.F., $F(x, y)$, obtained? Finding either the P.D.F. or the C.D.F. would be suitable, but I am unsure of how to approach it.
I'm very familiar with bivariate random variables when both components are continuous or discrete, but the mixed case is giving me trouble. In drawing some mild inspiration from a very dated post (Find the distribution ,when parameter is random), I made an attempt:
$F(x,y)=\mathbb P(X≤x,Y≤y)=\mathbb P(X≤x)·\mathbb P(Y≤y∣X≤x)$
I'm unsure if the last step is truly adequate, but I'm more broadly seeking a way of combining continuous and discrete random variables into a joint distribution. Any guidance is very much appreciated. Thanks!
You can write the marginal distributions from the joint distribution, but not the joint distribution from the marginals (there would be infinitely many possible joint distributions in general).
However, knowing $P(X\leq x|Y=y)$ for $y=-1,1$ is enough to characterize the joint distribution:
$$P(X\leq x, Y=y)=P(X\leq x|Y=y)P(Y=y).$$