Random Variable: Independent or Dependent?

Question

You are given two independent random variables X and Y, where

Pr(X=0) = Pr(X=1) = Pr(Y=0) = Pr(Y=1) = $\frac{1}{2}$

Define the random variable Z = X*Y

Are the random variables X and Z independent or dependent?

I know for the random variables to be independent, it should follow that:

P(X=x ^ Z=z) = P(X=x)*P(Z=z)

In this case, taking X and Z at 0:

Pr(X=0 ^ Z=0) = $\frac{1}{2}$ * $\frac{1}{2}$ = $\frac{1}{4}$

I calculated the probability of Z by considering there is a 2/4 probability of Z=0. Not confident about that.

Pr(X)*Pr(Z) = $\frac{1}{2}$ * $\frac{1}{4}$ = $\frac{1}{8}$

For Pr(Z) I got $\frac{1}{4}$ by multiplying the probabilities of Y and X at 0.

Since, $\frac{1}{4}$ not = $\frac{1}{8}$ they are not independent.

Did I approach this correctly?

Answer 1

Check the distribution of $Z$. First, it can only take the values $0$ and $1$. But as $\{Z=1\}$ is equivalent to $\{X=1\wedge Y=1\}$, then $P(Z=1)=\frac12\cdot\frac12=\frac14$ and so $P(Z=0)=\frac34$.

Also check that $$P(Z=0|X=0)=1,$$ and since $P(Z=0)\neq 1$, this implies $X$ and $Z$ are not independent, because knowing that $X=0$ modifies the probability of $Z=0$, that is $$P(Z=0|X=0)\neq P(Z=0).$$

Answer 2

Hint: $XY$ cannot be $1$ when $X$ is $0$. However $X$ may be $0$ and $XY$ may be $1$.