Free SOA Exam P (Probability) Continuous Distributions Practice Questions

Continuous distributions on SOA Exam P include uniform, exponential, normal, gamma, and beta. Questions test density functions, cumulative distribution functions, percentiles, and transformations of continuous random variables.

128 questions 57 easy 41 medium 30 hard 2026 syllabus

Sample Questions

Question 1 Easy
X ~ Beta(3, 2). Find E[X].
Solution
C is correct.

For Beta(α\alpha,β\beta): E[X]E[X] = α\alpha/(α\alpha+β\beta) = 33+2\frac{3}{3+2} = 35\frac{3}{5}.
Question 2 Medium
XX ~ Exponential(1). Find E[eX/2]E[e^{X/2}].
Solution
A is correct.

E[e(tX)]E[e^{(tX)}] =M(t)== M(t) = 11−t\frac{1}{1−t} for t<1t < 1. E[e(X/2)]E[e^{(X/2)}] =M(= M((1)2\frac{(1)}{2})=1/(1−) = 1/(1-(1)2\frac{(1)}{2})=2) = 2.
Question 3 Hard
X has PDF f(x) = 2/(1+x)³ for x > 0. What is E[X]?
Solution
E is correct.

Verify: ∫0∞2(1+x)3dx=2⋅[−12(1+x)2]0∞=2⋅12=1\int_0^\infty \frac{2}{(1+x)^3}dx = 2 \cdot \left[-\frac{1}{2(1+x)^2}\right]_0^\infty = 2 \cdot \frac{1}{2} = 1. ✓

E[X]=∫0∞2x(1+x)3dxE[X] = \int_0^\infty \frac{2x}{(1+x)^3}dx

Let u=1+xu = 1+x, du=dxdu = dx, x=u−1x = u-1:
=∫1∞2(u−1)u3du=2∫1∞(u−2−u−3)du=2[−u−1+u−22]1∞= \int_1^\infty \frac{2(u-1)}{u^3}du = 2\int_1^\infty (u^{-2} - u^{-3})du = 2\left[-u^{-1} + \frac{u^{-2}}{2}\right]_1^\infty
=2[(0−0)−(−1+1/2)]=2×1/2=1= 2\left[(0-0) - (-1 + 1/2)\right] = 2 \times 1/2 = 1

This is a Pareto II with α=2,θ=1\alpha = 2, \theta = 1: E[X]=θ/(α−1)=1/1=1E[X] = \theta/(\alpha-1) = 1/1 = 1.
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