Worked solution · Exam P

SOA Exam P Sample Question #709 Solution

This Exam P sample reference tests Continuous Random Variables. The unknown proportionality constant cancels when the upper-tail integral is divided by the integral over the full support. The ratio is ln(5)/ln(10) = 0.698970, which rounds to 0.70 and matches choice D.

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.

Read the solution Open official source

Independent solution

How to solve this Continuous Random Variables question

Setup

Represent the density by a positive constant times its stated kernel over the finite support.

f(x)=c\frac{x}{1+x^2},\qquad 0<x<3

Model

For a probability ratio, the same normalizing constant appears in the numerator and denominator and therefore cancels.

\Pr(X>1)=\frac{\int_1^3 x/(1+x^2)\,dx}{\int_0^3 x/(1+x^2)\,dx}

\int \frac{x}{1+x^2}\,dx=\frac{1}{2}\ln(1+x^2)

Compute

Evaluate the logarithmic antiderivative at the two sets of limits and simplify the ratio.

\Pr(X>1)=\frac{\tfrac12[\ln(10)-\ln(2)]}{\tfrac12[\ln(10)-\ln(1)]}

\Pr(X>1)=\frac{\ln(5)}{\ln(10)}=0.6989700043\ldots

Answer

The requested upper-tail probability rounds to 0.70.

\boxed{\Pr(X>1)\approx 0.70\quad\text{(D)}}

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.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

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

AThe value 0.15 is close to one-half of log base 10 of 2. It both keeps an antiderivative factor that should cancel and reports a lower-tail quantity instead of the requested upper tail.
BDropping the numerator x changes the kernel to 1/(1+x²); its normalized upper tail is about 0.3712, which points to B but solves a different density.
CReplacing the correct logarithmic ratio 10/2 = 5 by (3²-1)/(1+1²) = 4 gives log base 10 of 4 = 0.6021. The plus one in the antiderivative cannot be changed to a subtraction.
EKeeping an extra one-half in the lower-tail probability gives 0.5 log base 10 of 2 = 0.1505; taking its complement then gives 0.8495, near E, but the one-half cancels from the normalized ratio.

Original practice · fully worked

Original variant: acoustic signal intensity

A marine recorder assigns a dimensionless intensity score S between 0 and 4 to each detected acoustic event. The score has a continuous density proportional to s on that interval. Calculate the probability that a detected event has an intensity score greater than 3.

A
0.0625
B
0.2500
C
0.4375
D
0.5625
E
0.7500

Answer and calculation

Normalizing a density proportional to s gives f(s) = s/8. Its probability above 3 is the ratio (4 squared − 3 squared)/4 squared = 7/16 = 0.4375, so choice C is correct.

Setup

Introduce the normalizing constant for the linearly increasing density.

f(s)=cs,\qquad 0<s<4

Model

Choose c so that the total area under the density is one.

1=c\int_0^4 s\,ds=c\frac{4^2}{2}=8c

c=\frac{1}{8}

Compute

Integrate the normalized density over scores above 3.

\Pr(S>3)=\int_3^4\frac{s}{8}\,ds

\Pr(S>3)=\frac{4^2-3^2}{16}=\frac{7}{16}=0.4375

Answer

The upper-tail probability is 0.4375.

\boxed{\Pr(S>3)=0.4375\quad\text{(C)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P Univariate Random Variables practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Continuous Random Variables question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: acoustic signal intensity

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables