probability distributions

Question

Assume that we have assessed probability distributions for two discrete uncertain quantities: Variable 1 and Variable 2, and that they are probabilistically dependent (mutually relevant). How many of the following statements are true?
I. Knowing the value of Variable 1 may cause us to reassess the probability distribution for Variable 2
II. If we have assessed the conditional distributions for Variable 2 given Variable 1, we know that we can use that same probability distribution for Variable 1 given Variable 2
III. If the conditional probability distribution for Variable 2 is the same for every value of Variable 1, then we know that the variables are not probabilistically dependent.

Answer 1

Hi Steve!

I. The first question relates to the idea of dependence. If two variables are dependent, then knowing the value of one of them modifies your understanding of the second. So yes! knowing the value of Variable 1 may cause us to reassess the probability distribution for variable 2.

II. This relates to conditional distributions. While it's true that the two variables being dependent means that the distribution of variable 1 is not the same as the distribution of variable 1 given the value of variable 2, it is not the case that conditional distributions are identical. That is V1 given V2 is not the same as V2 given V1, in general.

III. Yes! If the conditional distribution of V2 is the same for every value of V1, then we're saying that the conditional distribution of V2 is the same as the marginal distribution of V2. That is, the distribution of V2 is the same as the distribution of V2|V1. This is one of the definitions of independence, so the two variables would not be dependent in this case.