I have $2$ random discrete variables $X$ and $Y$ with known probability $P[X = x]$ and $P[Y = y]$ and correlation coefficient $\rho$.
I would like to know how to compute the joint probability $P[X = x $ and $Y = y]$ taking into account the correlation, but without involving the conditional probability $P[Y = y | X = x]$.
Edit: I stumble upon this approach
One can use the definition of correlation coefficient as follows:
\begin{align}
\rho = \frac{P[X = x, Y = y] - P[X = x]P[Y = y]}{\sqrt{P[X = x](1 - P[X = x]) P[Y = y] (1 - P[Y = y])}}
\end{align}
Re-arrange the terms and solve for $P[X = x, Y = y]$.
Can anyone please tell me or direct me to a source where I can confirm the validity of this formula of $\rho$?
Any answer will be much appreciated!