Verified solution · Exam P

SOA Exam P Sample Question #246 Solution

This Exam P sample reference tests Poisson Distribution. Conditioning two independent Poisson counts on their sum makes X binomial with n=3 and p=1.7/4=0.425. Since X-Y=2X-3, the conditional variance is 4(3)(0.425)(0.575)=2.9325, choice C.

Independent analysis by ActuaryProof · reviewed August 4, 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 Poisson Distribution question

Setup

Condition on the fixed combined count and use the Poisson splitting property.

X\mid(X+Y=3)\sim\operatorname{Binomial}\left(3,\frac{1.7}{1.7+2.3}\right)

p=0.425

Model

Under the condition, eliminate Y from the difference using the fixed total.

Y=3-X,\qquad X-Y=2X-3

Compute

A constant does not affect variance, while a scale factor is squared.

\operatorname{Var}(X-Y\mid X+Y=3)=4\operatorname{Var}(X\mid X+Y=3)

\operatorname{Var}(X-Y\mid X+Y=3)=4(3)(0.425)(0.575)=2.9325

Answer

The conditional variance rounds to 2.93.

\boxed{2.93\quad\text{(C)}}

Continue without hunting through PDFs

All 718 Exam P sample solutions in syllabus order

The searchable 3108-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam P study manual PDF

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 1.71 is 3(0.57), obtained by rounding the conditional share 0.575 down before reporting E[Y|X+Y=3]. It is a component mean, not the variance of the difference.
BThe value 1.77 can result from miscopying p(1-p)=0.244375 as 0.1475 before evaluating 4(3)p(1-p). Using the actual conditional binomial parameter gives 2.9325.
DThe value 3.14 is E[(X-Y)²|X+Y=3]=Var(X-Y|X+Y=3)+E[X-Y|X+Y=3]²=2.9325+0.2025=3.135. It is the second raw moment, not the variance.
EThe value 4.00 is the unconditional variance Var(X-Y)=Var(X)+Var(Y)=1.7+2.3. Conditioning on the combined count changes the variance.

Original practice · fully worked

Original variant: conditional covariance from an inferred split

Counts A and B from two independent event streams are Poisson with unknown positive means. Given that six events occurred in total, the conditional probability that all six came from stream A is 1/64. Calculate Cov(A,B) under this condition.

A
-3.000
B
-1.500
C
0
D
1.500
E
6.000

Variant answer in brief

Conditioning on the total makes A binomial with six trials and an unknown split probability p. The all-A probability gives p=1/2, and because B=6-A, the conditional covariance is -Var(A)=-1.5, choice B.

Setup

Let p be stream A's share of the combined Poisson rate. Conditional splitting makes A binomial.

A\mid(A+B=6)\sim\operatorname{Binomial}(6,p)

Model

Use the probability that every conditioned event is allocated to stream A to recover p.

\Pr(A=6\mid A+B=6)=p^6=\frac1{64}

p=\frac12

Compute

Under the fixed total, B is a decreasing affine function of A.

B=6-A

\operatorname{Cov}(A,B\mid A+B=6)=\operatorname{Cov}(A,6-A)

=-\operatorname{Var}(A)=-6\left(\frac12\right)\left(\frac12\right)=-1.5

Answer

The conditional covariance is -1.500.

\boxed{-1.500\quad\text{(B)}}

Keep practicing

Turn this solution into a complete study sequence

The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

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

Cite this solution

ActuaryProof. “SOA Exam P Sample Question #246 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Poisson Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: conditional covariance from an inferred split

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution