Two random variables are independently distributed if their joint distribution is the product of their marginal distributions
It is intuitively easier to understand that two random variables are independently distributed if all conditional distributions of Y given X are equal. Derive one of the two conditions from the other.
What will be an ideal response?
Answer: If all conditional distributions of Y given X are equal, then
But if all conditional distributions are equal, then they must also equal the marginal distribution, i.e.,
Given the definition of the conditional distribution of Y given X = x, you then get
Pr(Y = y = x) = = Pr(Y = y),
which gives you the condition
Pr(Y = y, X = x) = Pr(Y = y) Pr(X = x).
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