Q:

Let $X=(X_{1},X_{2})$ be s.t. $X_{1}$ ~ $Bin(1,p)$, $X_{2}|X_{1}$ ~ $Bin(1,p)$ and $X_{2}|X_{1}$ ~ $Bin(1,\frac{1}{2})$,find the joint distribution

Let $X=(X_{1},X_{2})$ be bivariate random variable such that $X_{1}$ ~ $Bin(1,p)$, $X_{2}|X_{1}=1$ ~ $Bin(1,p)$ and $X_{2}|X_{1}=0$ ~ $Bin(1,\frac{1}{2})$
How can I find the joint distribution of $(X_{1},X_{2})$

A:

How can I find the joint distribution 

On a case by case basis.
$$\mathsf P(X_1{=}x, X_2{=}y) \;=\; \begin{cases} \mathsf P(X_1{=}0)\;\mathsf P(X_2{=}y\mid X_1{=}0) & : x=0, y\in\{0,1\} \\[1ex] \mathsf P(X_1{=}1)\;\mathsf P(X_2{=}y\mid X_1{=}1) & : x=1, y\in\{0,1\}\end{cases}$$

