Independent random variables are uncorrelated. In general, uncorrelated random variables need not be independent. But in specific examples they are.
Question: Let ξ and η be Bernoulli(p) and Bernoulli(r) random variables, 0 < p, r < 1. Show that if ξ and η are uncorrelated then they are independent.
My attempt: I'm not exactly sure where to begin with this. I know that if the covariance between two random variables is equal to zero, then they are said to be uncorrelated. However, I struggled to complete this.
To show that two discrete variables are independent, it suffices to show that $P(X=a\cap Y=b)=P(X=a)P(Y=b)$ for all values $a,b$ that $X,Y$ can take. Here, $a,b$ can both only be $0$ and $1$. I'll take care of the case $a=b=1$.
The correlation between $\xi$ and $\eta$ is equal to $\frac{E[\xi \eta]-E[\xi]E[\eta]}{\sqrt{\text{Var }\xi\text{ Var }\eta}}$. If this is zero, then this implies $$ E[\xi\eta]=E[\xi]E[\eta]=p\cdot r=P(\xi=1)P(\eta=1) $$
To find $E[\xi\eta]$, note that $\xi\eta$ is always equal to $0$ or $1$, so $$ E[\xi\eta] = P(\xi\eta=1)=P(\xi=1\cap \eta=1) $$
The above proves that $$P(\xi=1\cap \eta=1)=P(\xi=1)P(\eta=1).$$