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.

318 Questions

163 Easy

97 Medium

58 Hard

2026 Syllabus

Sample Questions

Question 1 Easy

Cov(X, 3X + 5) = ? given Var(X) = 4.

Solution

C is correct.

\text{Cov}(X, 3X+5)

= 3

\cdot \text{Var}(X)

+ 0 = 3(4) = 12.

Question 2 Medium

X, Y iid Uniform(0,1). Calculate P(X + Y < 1).

Solution

D is correct.

P(X+Y < 1)

= area of triangle below x+y=1 in unit square =

\frac{(1)}{2}

Question 3 Hard

Let

X \sim N(0,1)

and

Y = X^2

. Find

\text{Cov}(X, Y)

Solution

A is correct.

\text{Cov}(X, Y) = E[XY] - E[X] \cdot E[Y] = E[X^3] - E[X] \cdot E[X^2]

For

X \sim N(0,1)

, all odd moments are zero by symmetry:

E[X] = 0

E[X^3] = 0

. Also

E[X^2] = 1

\text{Cov}(X, X^2) = 0 - 0 \cdot 1 = 0

\text{Cov}(X, Y) = 0

does NOT imply independence. Here

Y = X^2

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 = X^2

is purely nonlinear.

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