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.

 

[CellCoree]

No, random variables based on the same probability mass function (pmf) or probability density function (pdf) are not always independent. Independence between random variables means that the occurrence of one event does not affect the probability of the other event occurring. This is not always the case for random variables with the same pmf or pdf.

For example, consider two random variables X and Y that represent the outcomes of rolling two dice. If X represents the outcome of the first die and Y represents the outcome of the second die, they both have the same pmf (uniform distribution) and are independent. However, if X represents the sum of the two dice and Y represents the difference between the two dice, they both have the same pmf (triangular distribution) but are not independent.

In terms of multiplying their expected values (E[X]) together, this only applies to independent random variables. For non-independent random variables, the expected value of their product is not equal to the product of their expected values. Therefore, it is not always possible to form another random variable with the same distribution and pmf or pdf by multiplying the expected values of two random variables together. Overall, independence between random variables is not guaranteed solely based on their pmf or pdf.