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

X takes values 1 and 2 and Y takes values 0 and 10. Joint probabilities: P(1,0)=0.30, P(1,10)=0.10, P(2,0)=0.20, P(2,10)=0.40. What is E[Y | X=1] and does it equal E[Y]?

E[Y given X=1] is 2.5 while E[Y] is 5, so they differ. Given X=1, Y equals 10 with probability 0.10/0.40 = 0.25, but unconditionally it equals 10 with probability 0.50. This shows X and Y are dependent.

  1. AE[Y|X=1]=2.5, and E[Y]=5, so they differCorrect
  2. BE[Y|X=1]=2.5, and E[Y]=2.5, so they are equal
  3. CE[Y|X=1]=1.0, and E[Y]=5, so they differ
  4. DE[Y|X=1]=5, and E[Y]=5, so they are equal

Explanation

P(X=1)=0.40. P(Y=10|X=1)=0.10/0.40=0.25, so E[Y|X=1]=2.5. P(Y=10)=0.10+0.40=0.50, so E[Y]=5. The unequal values show dependence; 1.0 is the unscaled 0.10×10.

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