Suppose three random variables X , Y , Z have a joint distribution.

Question

Suppose three random variables X , Y , Z have a joint distribution.

PU,PV,W(u,v,w)=toproaba gooas.P(not mov eveuy now)

Then, U and V are independent when in respect to Z at the exit was printron.

False

Answer 1

To determine whether U and V are independent with respect to Z, we need to examine the joint distribution of U, V, and Z.

Given the joint distribution PU,PV,W(u,v,w), the independence condition states that the joint distribution can be factored into the product of the marginal distributions:

PU,PV,W(u,v,w) = PU(u) × PV(v) × PZ(w)

In other words, if U and V are independent, the probability of U and V taking certain values should not depend on the value of Z.

From the given statement "PU,PV,W(u,v,w)=toproaba gooas.P(not mov eveuy now)", it is not clear whether the joint distribution can be factored into the product of the marginals. Therefore, we cannot conclude that U and V are independent with respect to Z based on the given information.

In summary, the statement "U and V are independent with respect to Z" is false based on the given information.

Answer 2

The statement is false. Two random variables, U and V, are independent if and only if their joint distribution can be expressed as the product of their marginal distributions. In this case, the joint distribution of U and V is given by PU,PV,W(u,v,w). If U and V were independent with respect to Z, the joint distribution would be the product of PU(u,w) and PV(v,w) for some marginal distributions PU(u,w) and PV(v,w). However, based on the information given, the joint distribution does not factorize in this way. Therefore, U and V are not independent with respect to Z.