Verified 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 August 4, 2026

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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)}}

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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

Variant answer in brief

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)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #616 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.

How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: hybrid calibration offset

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution