Worked solution · Exam P

SOA Exam P Sample Question #578 Solution

This Exam P sample reference tests Normal Distribution. The independent profit components sum to a normal variable with mean 600 and standard deviation √(400²+300²)=500. Standardizing zero gives -1.20, so the requested probability is P(Z>-1.20) and choice B is correct.

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 Normal Distribution question

Setup

Add the component means to obtain the center of overall profit.

E[S]=200+400=600

Model

The sum remains normal, and independence makes the variances additive.

\operatorname{Var}(S)=400^2+300^2=250000

S\sim N(600,500^2)

Compute

Standardize the zero-profit boundary.

\Pr(S>0)=\Pr\!\left(Z>\frac{0-600}{500}\right)

\Pr(S>0)=\Pr(Z>-1.20)=0.8849303298\ldots

Answer

The standardized expression matches choice B.

\boxed{\Pr(Z>-1.20)\quad\text{(B)}}

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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 uses an aggregate spread that is too small, producing an excessively large standardized mean magnitude.
CThis adds the component standard deviations to get 700, yielding 600/700 about 0.86 rather than using the root-sum-of-squares spread.
DThe displayed z-score 0.83 implies an aggregate standard deviation near 600/0.83=723, which is even larger than the unsupported arithmetic sum 400+300=700.
EThis standardizes the first profit component alone, using 200/400=0.50 and omitting the second component.

Original practice · fully worked

Original variant: inferred reading correlation

Two jointly normal calibration readings A and B have means 10 and 6 and standard deviations 3 and 4, respectively. The probability that A-B is positive is 0.8413447461. Calculate the correlation between A and B.

A
-0.375
B
0.000
C
0.375
D
0.750
E
1.000

Answer and calculation

The difference has mean 4, and the stated probability is Φ(1), so its standard deviation must be 4. Equating Var(A-B)=25-24rho to 16 gives rho=0.375, so choice C is correct.

Setup

Let D be the difference and express its mean and variance in terms of the unknown correlation.

D=A-B

E[D]=10-6=4

\operatorname{Var}(D)=3^2+4^2-2\rho(3)(4)=25-24\rho

Model

The stated positive probability corresponds to a standard-normal score of one.

\Pr(D>0)=\Phi\!\left(\frac{4}{\operatorname{SD}(D)}\right)=\Phi(1)

\operatorname{SD}(D)=4

Compute

Set the variance expression equal to 16 and solve for the correlation.

25-24\rho=16

\rho=\frac9{24}=0.375

Answer

The readings have correlation 0.375.

\boxed{\rho=0.375\quad\text{(C)}}

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How to solve this Normal Distribution question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: inferred reading correlation

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables