I'm struggling to find the joint distribution $P(X\le x, Y=0)$, where $X\sim$ Unif$[0,1]$ and $Y\sim$ Bernoulli$(X)$. I can compose
$$P(X\le x, Y=0)=P(X\le x\mid Y=0)\cdot P(Y=0)$$, but this is where I hit a snag. How do I solve $P(X\le x\mid Y=0)$ when I don't know $P(X≤x, Y=0)$, and the definition of continuous conditional distributions uses it? I think it's gotta be like $\int_0^x$ of something, but I don't know what to integrate. Any intuition here?
You are given a marginal density and a conditional pmf, that is
$$f_X(x)=\mathbb{1}_{[0;1]}(x)$$
and
$$p_{Y|X}(y|x)=x^y(1-x)^{1-y}$$
where $y=0;1$ and $x \in [0;1]$
thus the requested joint distribution is, using the definition
$$P(X\leq x;Y=0)=\int_0^x(1-t)dt=x-\frac{x^2}{2}$$