Free SOA Exam P (Probability) Continuous Distributions Practice Questions

Practice continuous distributions for Exam P, including uniform, exponential, normal, gamma, and beta. Questions test density functions, CDFs, percentiles, and transformations of continuous random variables.

98 Questions
55 Easy
25 Medium
18 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
X ~ Beta(3, 2). Find E[X].
Solution
For Beta(α\alpha,β\beta): E[X]E[X] = α\alpha/(α\alpha+β\beta) = 33+2\frac{3}{3+2} = (3)5\frac{(3)}{5}.
Question 2 Medium
X ~ Exponential(1). Find E[e^(X/2)].
Solution
E[e(tX)]E[e^{(tX)}] = M(t) = 11−t\frac{1}{1−t} for t < 1. E[e(X/2)]E[e^{(X/2)}] = M((1)2\frac{(1)}{2}) = 1/(1−(1)2\frac{(1)}{2}) = 2.
Question 3 Hard
X has PDF f(x) = 2/(1+x)³ for x > 0. What is E[X]?
Solution
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.

Distractor analysis:
- 0.500: Computes θ/(α)=1/2\theta/(\alpha) = 1/2.
- 1.500: Computes αθ/(α−1)=2\alpha\theta/(\alpha-1) = 2... no. Uses different Pareto.
- 2.000: Uses α=2\alpha = 2 directly.
- ∞: Thinks the integral diverges (it would if α≤1\alpha \le 1).

The answer is 1.0001.000.
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More Exam P Topics

Probability Fundamentals Conditional Probability & Bayes Discrete Distributions Joint & Marginal Distributions Expectation & Variance Insurance Applications

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