Conditional probability distribution of one variable given that it is equal to another variable

Question

If I have two random variables $X$ and $Y$ that map to the same sample space $S = \{1, 2, 3, ..., N\}$, and each variable has a probability distribution $\pi_x(x)$ and $\pi_y(y)$, then what is the conditional probability distribution of $X$ if we know that $X = Y$? My first thought was that it would just be $\pi_y(y)$, but it doesn't make sense to me that the conditional distribution would be independent of $\pi_x(x)$.

Answer 1

Indeed, you can see that your answer ought to be symmetric in $X$ and $Y$, since $P(X=x|X=Y)$ must be the same as $P(Y=x|X=Y)$.

Suppose you know $X$ and $Y$ are independent. Then \begin{align*} P(X=x|X=Y)=&\frac{P(X=x,X=Y)}{P(X=Y)}\\ =&\frac{P(X=x,Y=x)}{\sum_y P(X=y, Y=y)}\\ =&\frac{P(X=x)P(Y=x)}{\sum_y P(X=y)P(Y=y)}\\ =&\frac{\pi_X(x)\pi_Y(x)}{\sum_y \pi_X(y)\pi_Y(y)}. \end{align*} So the mass at $x$ is proportional to $\pi_X(x)\pi_Y(x)$.

If you don't assume independence, you need to know more about the joint probability mass function of $X$ and $Y$. Let $\pi_{X,Y}(x,y)=P(X=x, Y=y)$. Then in a similar way $$ P(X=x|X=Y)=\frac{\pi_{X,Y}(x,x)}{\sum_y \pi_{X,Y}(y,y)}. $$