Are random variables based on the same pmf or pdf always independent?

[EdmureTully]

Are they always independent from each other so that you can multiply their E[X] together to form another E[X] with the same distribution and pmf or pdf?

 

[jbunniii]

No, in general they need not be independent. In the most extreme case, you may have something like X = some random variable, and Y = X. Then X and Y are 100% correlated, so they are certainly not independent.

 

[EdmureTully]

How do you verify their independence then? Any quick and easy way? Any obvious cases?

 

[chiro]

Hey EdmureTully.

The condition for independence between random variables is P(A = a, B = b) = P(A=a)*P(B=b) where the left hand side is the joint distribution.

If you have this separability, then the two random variables are by definition independent.