This Exam P sample reference tests Poisson Distribution. The payment is 0, 1000, or 2000 according as the Poisson count is 0, 1, or at least 2. Its first two moments give standard deviation about 699, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
DThe value 817 is the rounded second moment 816892.5 expressed in thousands. A second moment is not a standard deviation.
EThe value 904 is approximately the square root of the uncentered second moment. It omits subtraction of the squared mean.
Original practice · fully worked
Original variant: service credits capped after two outages
A cloud service experiences N outages in a week, where N is binomial with three trials and outage probability 0.20. It grants an 80-dollar credit per outage but never more than 160 dollars. Find the standard deviation of the weekly credit.
C 44.80 dollars
B 50.60 dollars
A 53.64 dollars
D 58.24 dollars
E 64.00 dollars
Variant answer in brief
The credit states 0, 80, and 160 have probabilities 0.512, 0.384, and 0.104. Their moments give a standard deviation of about 53.64 dollars.
Setup
Setup
Capping three-outage credits at 160 creates payment states 0, 80, and 160 from a Binomial(3,0.2) count.
N∼Bin(3,0.2)
C=80min(N,2)
Model
Model
Compute the first two payment-state probabilities directly; the remaining probability 0.104 belongs to two or three outages and hence to the cap.
P(C=0)=0.83=0.512
P(C=80)=3(0.2)(0.8)2=0.384
P(C=160)=0.104
Compute
Compute
The credit has mean 47.36 and second moment 5120, so its centered standard deviation is 53.64 dollars.
E[C]=47.36
E[C2]=5120
SD(C)=5120−47.362=53.64
Answer
Answer
Therefore the weekly-credit standard deviation is 53.64 dollars, corresponding to choice A.
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