IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
Let N be Poisson with mean 4, and given N = n, X is binomial with n trials and probability 0.25. What is Var(X)?
Var(X) is 1.0, using the conditional variance formula: expected conditional variance 0.75 plus variance of the conditional mean 0.25.
- A0.75
- B1.0Correct
- C1.25
- D3.0
- 4.0
Explanation
E(X|N) = 0.25N and Var(X|N) = N(0.25)(0.75) = 0.1875N. E[Var(X|N)] = 0.1875*4 = 0.75. Var[E(X|N)] = 0.0625*4 = 0.25. Total = 0.75 + 0.25 = 1.0. Using only the first term gives 0.75.
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