Let $X, Y$ and $Z$ are all binary random variables, and $Y = X + Z \mod 2.$
$X$ and $Z$ are independent.
a) Suppose $X$ and $Z$ are both uniformly distributed. Are $X$ and $Y$ independent? Why? (Prove or disprove it.)
b) Suppose $X$ and $Z$ are not uniformly distributed. Are $X$ and $Y$ independent? Why? (Prove or disprove it.)
What I have tried for part (a) since $X$ and $Z$ are independent and uniformly distributed over $\{0,1\}$ then $P(X=0) =P(X=0,Z=0) + P(X=1,Z=1) = .5\times .5 +.5\times .5 = .5$ then $p(Y=1) =.5$ in case (a) $Y$ also is uniformly distributed. $P(X=0,Y=0) = P(X=0) P(Y=0|X=0)$ how can I complete the proof for part (a) I don't know how to calculate $P(Y=0|X=0)$ ?
How about part (b) ?
Use the formula for conditional probability: $$P(Y=0 \mid X=0) = \frac{P(Y=0 \text{ and } X=0)}{P(X=0)}$$ and re-write the numerator as a probability involving only $X$ and $Z$. Then use independence of $X$ and $Z$.