Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, (Y,Z) is uniformly distributed on the set S={(0,0),(0,2),(2,0),(2,2)}. The figure below shows all the possible values for the triple (X,Y,Z) that have X≤8. (Note that the X axis starts at 1 and that a complete figure would extend indefinitely to the right.)
Answer the following with “Yes" or “No":
a) Are Y and Z independent?
- unanswered
b)Given that Z=2, are X and Y independent?
- unanswered
c) Given that Z=0, are X and Y independent?
- unanswered
d) Given that Z=2, are X and Z independent?
a) no
b) yes
c) no
d) yes
Why yes for the last one?
To determine whether Y and Z are independent, we need to check if the joint probability distribution of Y and Z can be factored into the product of their marginal distributions.
In this case, when X is even, Y and Z are both equal to zero. Therefore, the joint probability distribution of Y and Z is:
P(Y=0, Z=0) = P(X=even) * P(Y=0 | X=even) * P(Z=0 | X=even)
Since Y and Z are both zero when X is even, this joint probability is equal to 1.
Now, let's consider the case when X is odd. In this case, (Y,Z) is uniformly distributed on the set S={(0,0),(0,2),(2,0),(2,2)}. The joint probability distribution of Y and Z in this case is:
P(Y=0, Z=0) = P(X=odd) * P(Y=0 | X=odd) * P(Z=0 | X=odd)
Since the events of Y=0 and Z=0 are not independent for X=odd (as Z=0 implies that Y cannot take the value 2), the joint probability distribution cannot be factored into the product of their marginal distributions. Hence, Y and Z are not independent.
Therefore, the answer to part a) is "No".
To answer part b), we need to determine if X and Y are independent given that Z=2. Given Z=2, we know that (Y,Z) can only take the values (0,2) and (2,2). In both of these cases, X can take the values 1, 3, 5, and so on.
To check for independence, we need to compare the joint distribution P(X,Y | Z=2) with the product of the marginal distributions P(X | Z=2) and P(Y | Z=2).
Since the joint probability distribution P(X,Y | Z=2) depends on the value of Z, we cannot conclude about the independence of X and Y given Z=2 without knowing the relationship between X and Y for each possible value of Z.
Therefore, the answer to part b) is "Unanswered".
Similarly, in order to answer part c), we need to determine if X and Y are independent given that Z=0. Given Z=0, we know that (Y,Z) can only take the value (0,0). In this case, X can take the values 2, 4, 6, and so on.
Again, to check for independence, we need to compare the joint distribution P(X,Y | Z=0) with the product of the marginal distributions P(X | Z=0) and P(Y | Z=0).
Since the joint probability distribution P(X,Y | Z=0) depends on the value of Z, we cannot conclude about the independence of X and Y given Z=0 without knowing the relationship between X and Y for each possible value of Z.
Therefore, the answer to part c) is "Unanswered".
Finally, to answer part d), we need to determine if X and Z are independent given that Z=2. Given Z=2, we know that (Y,Z) can only take the values (0,2) and (2,2). In both of these cases, X can take the values 1, 3, 5, and so on.
To check for independence, we need to compare the joint distribution P(X,Z | Z=2) with the product of the marginal distributions P(X | Z=2) and P(Z | Z=2).
Since the joint probability distribution P(X,Z | Z=2) can be factored into the product of the marginal distributions P(X | Z=2) and P(Z | Z=2), X and Z are independent given Z=2.
Therefore, the answer to part d) is "Yes".