Worked solution · Exam P

SOA Exam P Sample Question #165 Solution

This Exam P sample reference tests Normal Distribution. The variance factor 2.25 corresponds to a standard-deviation factor 1.5. A common value therefore has standardized score z_0.14/1.5 under the wider distribution, giving percentile 23.57%, choice D.

Independent analysis by ActuaryProof · reviewed July 31, 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 Normal Distribution question

Setup

Translate the variance relationship into the scale relationship used by normal quantiles.

\sigma_B^2=2.25\sigma_A^2

\sigma_B=1.5\sigma_A

Model

The two distributions share a mean, so equate the two representations of the common percentile value.

\mu+\sigma_A z_{0.14}=\mu+\sigma_B z_p

z_p=\frac{z_{0.14}}{1.5}

Compute

Evaluate the lower-tail score and map the rescaled score back through the normal CDF.

z_{0.14}=-1.0803193

z_p=-0.7202129

p=\Phi(-0.7202129)=0.2356970

Answer

The matching percentile for the wider distribution is approximately 23.6.

\boxed{23.6\%\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.

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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.

AThe value 5.3% comes from multiplying z_0.14 by 1.5 instead of dividing. A wider distribution makes the same below-mean value less extreme in standardized units.
BThe value 9.3% divides the percentile number itself by 1.5, computing 14%/1.5. Standard deviations rescale z-scores, not percentile probabilities.
CThe value 21.0% multiplies the percentile number itself by 1.5, computing 14%(1.5). The normal CDF must be applied after rescaling the z-score.
EThe value 31.6% results from dividing z_0.14 by 2.25, using the variance ratio where the standard-deviation ratio is required.

Original practice · fully worked

Original variant: infer a variance ratio from two alarm rates

Two automated lines produce measurement errors centered at a shared value m. Each error law is Gaussian, but the spreads may differ. At one fixed trigger h greater than m, line A crosses h on 2.5% of runs, whereas line B crosses h on 10% of runs. Find the quotient of B's variance and A's variance.

A
0.4275
B
0.6540
C
1.0000
D
1.5294
E
2.3390

Answer and calculation

The common alarm distance equals 1.95996 standard deviations for A and 1.28155 for B. Their standard-deviation ratio is 1.52937, so the variance ratio is 2.33897, choice E.

Setup

Let h be the shared alarm level and convert each upper-tail probability to a positive z-score.

z_A=\Phi^{-1}(0.975)=1.959964

z_B=\Phi^{-1}(0.90)=1.281552

Model

Both standardized distances represent the same physical distance h-mu.

h-\mu=z_A\sigma_A=z_B\sigma_B

Compute

Find the scale ratio and square it to obtain the requested variance ratio.

\frac{\sigma_B}{\sigma_A}=\frac{1.959964}{1.281552}=1.529368

\frac{\operatorname{Var}(B)}{\operatorname{Var}(A)}=(1.529368)^2=2.338967

Answer

Process B has about 2.339 times the variance of process A.

\boxed{2.3390\quad\text{(E)}}

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How to solve this Normal Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: infer a variance ratio from two alarm rates

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables