Let $X\sim b(1,p)$ and $Y\sim b(1,q)$, in other words, $X$ and $Y$ are in Bernoulli distribution.
Question: If $X$ and $Y$ are uncorrelated, does it imply $X$ and $Y$ are independent?
These are my steps:
$E(X)=p$ and $E(Y)=q$
Since $X$ and $Y$ are uncorrelated
$\begin{align}
Cov(X,Y)&=0\\
E(XY)-E(X)E(Y)&=0\\
E(XY)&=pq\\
\sum_{x} \sum_{y}\ xy \cdot p(x,y) &= pq\ \\
p(1,1) &= pq
\end{align}$
Since $P_{X}(1)=p$, then
$\begin{align}
P(1,0)+P(1,1) &= p\\
P(1,0) &= p-qp \\
&= p(1-q)
\end{align}$
For the same reason, $P(0,1)=q(1-p)$
Then $P(0,0)=1-pq-p(1-q)-q(1-p)=(1-p)(1-q)$
Since $p(1,1)$ is known, the remaining 3 probabilities $p(x,y)$ is also determined and equal to $p_{X}(x) \cdot p_{Y}(y)$
Therefore, in this case, $X$ and $Y$ are uncorrelated and independent.
Am I right for this proof?