I was refreshing my knowledge regarding the definition of joint probabilities and read a page from the book: 'Econometrics for dummies' which gave the following example.
What I intended to do was to calculate the joint probabilities $P(Y,X)$ from the unconditional probabilities $P(Y)$ and $P(X)$.
I came across the following two definitions of a joint probability:
Definition 1: $P(Y,X) = P(Y) \times P(X)$
Definition 2: $ P(Y,X) = P(Y|X) \times P(X) = P(X|Y) \times P(Y) $
Using Definition 1, I could not obtain the joint probabilities from the tables. For example multiplying $P(Y=1)=0.35$ and $P(X=2)=0.10$ did not give me $P(Y=1,X=2) = P(1,2) = 0$.
I do not understand why Definition 1 does not give me the right answer.
I believe the issue is that I am facing is:
I know that the conditional expectation of $P(Y|X)$ should not necesarily be equal to $P(Y)$. But does the second definition not imply that $P(Y|X)$ and $P(Y)$ are equal?