FRM Part I · FRM Exam Part I · Multivariate Random Variables
X and Y are bivariate normal with E[X]=5%, E[Y]=8%, σX=4%, σY=10%, and correlation 0.4. What is the conditional expected value of Y given X = 9%?
The conditional expectation is 12%. Using E[Y|X=x] = μY + ρ(σY/σX)(x − μX), the slope is 0.4 × 10/4 = 1.0, and X exceeds its mean by 4 percentage points, so Y's expected value rises from 8% to 12%.
- A8.0%
- B10.0%
- C12.0%
- D9.6%Correct
Explanation
E[Y|X] = μY + ρ(σY/σX)(x − μX) = 8% + 0.4(10/4)(4%) = 8% + 4% = 12%? Compute: 0.4 × 2.5 = 1.0; 1.0 × 4% = 4%; total 12%. So the correct value is 12%.
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