FRM Part I · FRM Exam Part I · Multivariate Random Variables
Two discrete variables X and Y have the joint pmf: P(0,0)=0.20, P(0,1)=0.20, P(1,0)=0.30, P(1,1)=0.30. Which statement is correct?
X and Y are independent. The conditional probability of Y=1 is 0.5 whether X is 0 or 1, equal to the marginal P(Y=1)=0.5, so every joint probability equals the product of marginals. Unequal joint cell probabilities alone do not indicate dependence.
- AX and Y are independent because P(Y=1|X=0)=P(Y=1|X=1)=0.5Correct
- BX and Y are dependent because the joint probabilities are not all equal
- CX and Y are independent because the marginal of X is uniform
- DX and Y are dependent because P(0,0) differs from P(1,0)
Explanation
P(X=0)=0.40, P(X=1)=0.60. P(Y=1|X=0)=0.20/0.40=0.5 and P(Y=1|X=1)=0.30/0.60=0.5, equal to P(Y=1)=0.5, so the joint equals the product of marginals in every cell. Unequal joint cells do not imply dependence.
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