FRM Part I · FRM Exam Part I · Multivariate Random Variables
Random variables X and Y have the joint distribution P(X=0,Y=0)=0.12, P(0,1)=0.28, P(1,0)=0.18, P(1,1)=0.42. Which statement is correct?
X and Y are independent. The conditional probability of Y=1 is 0.28/0.40 = 0.70 given X=0 and 0.42/0.60 = 0.70 given X=1, which equals the marginal probability of Y=1. When conditioning does not change the distribution, the variables are independent.
- AX and Y are independent because P(Y=1|X=0) = P(Y=1|X=1) = 0.70Correct
- BX and Y are dependent because P(X=1,Y=1) is not equal to 0.50
- CX and Y are independent because their covariance is positive
- DX and Y are dependent because P(Y=1|X=1) exceeds P(Y=1|X=0)
Explanation
P(X=0)=0.40, so P(Y=1|X=0)=0.28/0.40=0.70. P(X=1)=0.60, so P(Y=1|X=1)=0.42/0.60=0.70. The conditional probabilities equal the marginal P(Y=1)=0.70, so the variables are independent. Positive covariance would indicate dependence, not independence.
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