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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

X and Y have joint density f(x,y) = 2 for 0 < y < x < 1, and 0 otherwise. What is Var(Y | X = x) for 0 < x < 1?

Var(Y | X=x) = x²/12. Dividing the joint density 2 by the marginal 2x gives a conditional density 1/x on (0,x), which is uniform on (0,x), whose variance is x²/12.

  1. Ax²/12Correct
  2. Bx²/4
  3. Cx/12
  4. Dx²/3
  5. 1/12

Explanation

The marginal of X is f(x)=2x, so Y|X=x has density 2/(2x)=1/x on (0,x), i.e. uniform(0,x). Its variance is x²/12. The option 1/12 wrongly uses a uniform on (0,1).

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