In the proof of the Law of Total Expectation, the following identity confuses me
$$P(X=x|Y)P(Y=y)=P(X=x,Y=y)$$
I this the conditional probability formula? Is $P(X=x,Y=y)=P(X=x \cap Y=y)$?
2026-03-25 21:47:26.1774475246
Law of total expectation derivation
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Yes and yes, although for clarity $|Y$ should read $|Y=y$.