This is a correlation calculation for two binary random variables. The joint table gives covariance 0.0175 and marginal variances 0.1275 and 0.2475, so the correlation is approximately 0.098513 and rounds to choice B.
Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.
This is a correlation calculation for two binary random variables. The joint table gives covariance 0.0175 and marginal variances 0.1275 and 0.2475, so the correlation is approximately 0.098513 and rounds to choice B.
Setup
Setup
Read the two Bernoulli success probabilities and the joint-success probability from the joint distribution.
Pr(X=1)=0.05+0.10=0.15
Pr(Y=1)=0.45+0.10=0.55
E[XY]=Pr(X=1,Y=1)=0.10
Model
Model
For a Bernoulli variable, the variance is p(1-p). Center the product moment to obtain the covariance.
Var(X)=0.15(0.85)=0.1275
Var(Y)=0.55(0.45)=0.2475
Cov(X,Y)=0.10−(0.15)(0.55)=0.0175
Compute
Compute
Divide the covariance by the product of the two marginal standard deviations.
ρX,Y=(0.1275)(0.2475)0.0175
ρX,Y=0.0985134105…
Answer
Answer
The correlation rounds to three decimals as 0.099.
ρX,Y≈0.099(B)
Error diagnosis
Why the other letters tempt
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe covariance is 0.0175, which rounds to 0.018. Reporting covariance without standardizing it produces choice A.
CUsing the uncentered product moment gives 0.10 divided by sqrt(0.15 times 0.55), approximately 0.348; correlation requires centered moments.
DDividing covariance by the product of the variances, without taking the square root, gives approximately 0.555 and selects D.
EApplying one square root to the entire erroneous ratio Cov(X,Y)/(Var(X)Var(Y)) gives approximately 0.745 rather than the correlation formula.
Original practice · fully worked
Original variant: observatory maintenance signals
On a randomly selected night, let A indicate that an observatory's tracking motor needs recalibration and let B indicate that its cooling monitor sends an alert. Historical frequencies are 0.54 for neither event, 0.08 for recalibration only, 0.18 for an alert only, and 0.20 for both events. Calculate the correlation between A and B.
A 0.094
B 0.429
C 0.613
D 1.971
E -0.429
Variant answer in brief
The marginal event probabilities are 0.28 and 0.38, while the joint-event probability is 0.20. Standardizing covariance 0.0936 gives correlation 0.429480, so choice B is correct.
Setup
Setup
Form the two marginal event probabilities from the four operating states.
Pr(A=1)=0.08+0.20=0.28
Pr(B=1)=0.18+0.20=0.38
E[AB]=0.20
Model
Model
Use Bernoulli variances and subtract the product of the marginal means from the joint moment.
Var(A)=0.28(0.72)=0.2016
Var(B)=0.38(0.62)=0.2356
Cov(A,B)=0.20−(0.28)(0.38)=0.0936
Compute
Compute
Standardize the covariance by the two marginal standard deviations.
ρA,B=(0.2016)(0.2356)0.0936
ρA,B=0.4294799974…
Answer
Answer
The positive association between the two maintenance signals has correlation approximately 0.429.
ρA,B≈0.429(B)
Continue by topic
Build a complete proof notebook
The Probability Proof Manual reorganizes verified solutions by syllabus skill and adds formula proofs, error patterns, and original practice.