$E_1, E_2$ are independent events such that $P(E_1)=\frac14, P(E_2| E_1)=\frac12, P(E_1|E_2)=\frac14$. Define random variables $X$ and $Y$ by
$X=\begin{cases} & 1 & \text{if } E_1 \text{ occurs} \\ & 0 & \text{if } E_1^C \text{ occurs} \\ \end{cases}$
$Y=\begin{cases} & 1 & \text{if } E_2 \text{ occurs} \\ & 0 & \text{if } E_2^C \text{ occurs} \\ \end{cases}$
Consider the following statements:
$\alpha: X$ is uniformly distributed on the set [0,1]
$\beta: X,Y$ are identically distributed.
$\gamma: P(X^2+Y^2=1)=\frac12$
$\delta: P(XY=X^2Y^2)=1$
Choose the correct combination:
- $(\alpha,\beta)$ 2. $(\alpha,\gamma)$ 3. $(\beta,\gamma)$ 4. $(\gamma, \delta)$
Given answer is 4.
My attempt:
$P(E_2| E_1)=\frac12 \implies P(E_1 \cap E_2) = \frac18 $
$P(E_1|E_2)=\frac14 \implies P(E_2) = \frac12$
So that $P(X=1)=\frac14, P(X=0)=\frac34, P(Y=1)=\frac12, P(Y=0)=\frac12$
Now $P(XY=X^2Y^2)=P(XY(1-XY)=0)$
$=P(XY=0)+P(XY=1)$
$=P(X=0)P(Y=0)+P(X=1)P(Y=1)$
$=\frac34 \frac12+ \frac14 \frac12=\frac12$
This is where I'm lost. Please help!