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FRM Part I · FRM Exam Part I · Multivariate Random Variables

A risk analyst models the joint pmf of X (values 1, 2) and Y (values 0, 10) as: P(1,0)=0.30, P(1,10)=0.20, P(2,0)=0.10, P(2,10)=0.40. What is E[Y | X=2]?

E[Y|X=2] equals 8.0. Given X=2, the conditional probabilities are 0.2 for Y=0 and 0.8 for Y=10, obtained by dividing the joint probabilities by P(X=2)=0.50. The expectation is 10×0.8=8. Skipping normalization gives 4.0.

  1. A4.0
  2. B5.0
  3. C8.0Correct
  4. D6.0

Explanation

P(X=2)=0.10+0.40=0.50. Conditional probabilities: P(Y=10|X=2)=0.40/0.50=0.8, P(Y=0|X=2)=0.2. E[Y|X=2]=10×0.8=8.0. The 4.0 option uses the unnormalized joint 0.40×10, and 6.0 is the unconditional E[Y]=10×0.60.

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