I've found a good answer to a particular question I was given, though I'd like a second opinion on whether or not my answer is solid (I might be "begging the question" in this).

Show that two Bernoulli random variables $\displaystyle X$ and $\displaystyle Y$ are independent if and only if $\displaystyle P(X=1,Y=1)=P(X=1)P(Y=1)$.

Here's my answer.

Two Bernoulli random variables are independent if and only if they are uncorrelated, and thus have a covariance of zero ($\displaystyle Cov(X,Y)=0$).

Let $\displaystyle p_x$ be the pmf of $\displaystyle X$, and let $\displaystyle p_y$ be the pmf of $\displaystyle Y$.

If $\displaystyle X$ and $\displaystyle Y$ are independent then, by definition,

$\displaystyle Cov(X,Y)=p_{xy}-p_xp_y=P(X=1,Y=1)-P(X=1)P(Y=1)=0$,

as $\displaystyle P(X=1,Y=1)=P(X=1)P(Y=1)$.

If on the other hand we have that $\displaystyle Cov(X,Y)=0$, then

$\displaystyle p_{xy}-p_xp_y=0\Rightarrow p_{xy}=p_xp_y$.

Therefore, $\displaystyle X$ and $\displaystyle Y$ are independent.

Are there any mess-ups I might have made somewhere? I got this answer from this pdf file.