SOA Exam P Sample Question #654. Official problem wording is withheld; follow the source link to read it.

Verified solution · Exam P

SOA Exam P Sample Question #654

For three independent Uniform(0,10) observations, the maximum has distribution function (x/10)³. Integrating its density gives expected maximum 3(10)/4 = 7.5, which is choice B.

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Independent solution

How to solve this Order Statistics question

Answer in brief

For three independent Uniform(0,10) observations, the maximum has distribution function (x/10)³. Integrating its density gives expected maximum 3(10)/4 = 7.5, which is choice B.

Setup

Let M be the largest of the three independent uniform observations.

M=\max(X_1,X_2,X_3),\qquad X_i\sim\operatorname{Uniform}(0,10)

Model

The maximum does not exceed x exactly when all three observations do not exceed x.

F_M(x)=\Pr(X_1\le x,X_2\le x,X_3\le x)=\left(\frac{x}{10}\right)^3

f_M(x)=\frac{3x^2}{10^3},\qquad 0\le x\le10

Compute

Integrate x against the maximum density.

\operatorname{E}[M]=\int_0^{10}x\frac{3x^2}{10^3}\,dx

\operatorname{E}[M]=\frac{3}{1000}\left[\frac{x^4}{4}\right]_0^{10}=\frac{30}{4}=7.5

Answer

The expected largest observation is 7.5.

\boxed{7.5\quad\text{(B)}}

Error diagnosis

Why the other letters tempt

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

AThis is approximately the expected maximum of only two Uniform(0,10) observations, 20/3.
CThis is the expected maximum for four observations, not three.
DThis is approximately the five-observation value 50/6 and therefore assumes too many observations.
EThis places still more mass near the upper endpoint than a three-observation maximum can have.

Original practice · fully worked

Original variant: strongest calibration reading

Four independent calibration readings are uniformly distributed between 60 and 90. A technician records only the largest reading. Calculate its expected value.

A
75.0
B
80.0
C
82.5
D
84.0
E
85.0

Variant answer in brief

After scaling to Uniform(0,1), the maximum of four observations has mean 4/5. Mapping back to [60,90] gives 60 + 30(4/5) = 84, so choice D.

Setup

Standardize each reading to a unit-uniform variable.

U_i=\frac{X_i-60}{30}\sim\operatorname{Uniform}(0,1)

X_{(4)}=60+30U_{(4)}

Model

The maximum of four unit-uniform observations has density four times u cubed.

F_{U_{(4)}}(u)=u^4,\qquad f_{U_{(4)}}(u)=4u^3

Compute

Find the standardized expected maximum and transform it back to the original scale.

\operatorname{E}[U_{(4)}]=\int_0^1u(4u^3)\,du=\frac45

\operatorname{E}[X_{(4)}]=60+30\left(\frac45\right)=84

Answer

The expected strongest calibration reading is 84.

\boxed{84.0\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #654 — Verified Solution.” Verified 2026-07-30.

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