Free SOA Exam P (Probability) Joint & Marginal Distributions Practice Questions
Joint and marginal distributions are one of the most heavily weighted areas on SOA Exam P. Questions involve multivariate densities, marginal and conditional distributions, covariance, correlation, and tests of independence.
P(X+Y<1) = area of triangle below x+y=1 in unit square = 2(1).
Question 3
Hard
Let X∼N(0,1) and Y=X2. Find Cov(X,Y).
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Correct Answer: B
Solution
B is correct.
Cov(X,Y)=E[XY]−E[X]⋅E[Y]=E[X3]−E[X]⋅E[X2] For X∼N(0,1), all odd moments are zero by symmetry: E[X]=0, E[X3]=0. Also E[X2]=1. Cov(X,X2)=0−0⋅1=0 Cov(X,Y)=0 does NOT imply independence. Here Y=X2 is a deterministic function of X, so X and Y are completely dependent despite being uncorrelated. covariance only captures linear relationships, and the relationship Y=X2 is purely nonlinear.
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