Worked solution · Exam P

SOA Exam P Sample Question #616 Solution

This Exam P sample reference tests Uniform Distribution. The continuous uniform standard deviation is 10/√(12)=2.886751. The discrete uniform distribution on eleven consecutive integers has variance 10 and standard deviation 3.162278. Their absolute difference is 0.275526, which rounds to choice C.

Independent analysis by ActuaryProof · reviewed July 30, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Uniform Distribution question

Setup

Calculate the variance of the continuous uniform selection from its interval width.

\operatorname{Var}(C)=\frac{(10-0)^2}{12}=\frac{25}{3}

\operatorname{SD}(C)=\frac{10}{\sqrt{12}}=2.8867513459\ldots

Model

For the eleven equally likely integers, use the first two discrete moments.

E[D]=\frac{0+1+\cdots+10}{11}=5

E[D^2]=\frac{0^2+1^2+\cdots+10^2}{11}=35

Compute

Convert the discrete moments to a standard deviation and compare the two spreads.

\operatorname{Var}(D)=35-5^2=10

\operatorname{SD}(D)=\sqrt{10}=3.1622776602\ldots

\left|\operatorname{SD}(D)-\operatorname{SD}(C)\right|=0.2755263142\ldots

Answer

The absolute difference rounds to 0.28.

\boxed{0.28\quad\text{(C)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis assumes equal endpoints and equal means force the continuous and discrete uniform distributions to have the same spread.
BThis effectively uses only ten discrete values by dropping one endpoint; the resulting standard-deviation difference is about 0.02.
DThis applies a half-unit continuity adjustment directly to a standard deviation instead of recomputing the discrete variance.
EThis subtracts the variances, 10-25/3=1.6667, rather than subtracting the standard deviations.

Original practice · fully worked

Original variant: hybrid calibration offset

A calibration offset X is generated in one of two ways. With probability 0.60 it is continuous uniform on [-2,2]. Otherwise it is selected uniformly from the three values -3, 0, and 3. Calculate the unconditional standard deviation of X.

A
1.155
B
1.673
C
1.789
D
2.449
E
3.200

Answer and calculation

Both mixture components have mean zero. Their second moments are 4/3 and 6, so the unconditional variance is 0.60(4/3)+0.40(6)=3.2. Its square root is 1.788854, which rounds to choice C.

Setup

Both component distributions are symmetric about zero, so the mixture mean is zero.

E[X\mid M=1]=E[X\mid M=2]=0

E[X]=0

Model

Calculate the conditional second moments for the continuous and discrete components.

E[X^2\mid M=1]=\frac{(2-(-2))^2}{12}=\frac43

E[X^2\mid M=2]=\frac{(-3)^2+0^2+3^2}{3}=6

Compute

Average the component second moments, which equal their variances because both conditional means are zero.

\operatorname{Var}(X)=0.60\left(\frac43\right)+0.40(6)=3.2

\operatorname{SD}(X)=\sqrt{3.2}=1.7888543820\ldots

Answer

The unconditional standard deviation rounds to 1.789.

\boxed{1.789\quad\text{(C)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

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How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: hybrid calibration offset

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables