This Exam P sample reference tests Raw Moments of Rounded Variables. The continuous fourth moment is 2000, whereas rounding creates endpoint masses 1/20 and interior masses 1/10, giving fourth moment 2033.3 and difference 33.3, choice B.
How to solve this Raw Moments of Rounded Variables question
Setup
Setup
Compute the continuous fourth moment directly from the uniform density. Nearest-integer rounding gives the two endpoint cells half the probability of each interior cell.
E[X4]=101∫010x4dx=2000
Model
Model
Use probability one twentieth at the endpoints and one tenth at each interior integer to form the fourth raw moment of the rounded variable.
Pr(Y=0)=Pr(Y=10)=201,Pr(Y=k)=101(1≤k≤9)
Compute
Compute
The continuous and rounded fourth moments are 2,000 and 2,033.3. Their absolute difference is 33.3.
The absolute fourth-moment difference rounds to 33.
33(B)
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.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0 assumes nearest-integer rounding preserves the fourth raw moment. Rounding raises the moment from 2,000 to 2,033.3.
Original practice · fully worked
Original variant: second moment after rounding a reading
A sensor reading X is uniform on [0,4]. The reported value Y is X rounded to the nearest integer from 0 through 4. Find the absolute difference between the second raw moments of X and Y.
A 0
B 1/6
C 1/4
D 1/2
E 2/3
Answer and calculation
The continuous second moment is 16/3. The rounded second moment is 11/2, so the absolute difference is 1/6, choice B.
Setup
Setup
Compute the continuous second moment by integrating against the Uniform(0,4) density.
E[X2]=41∫04x2dx=316
Model
Model
Nearest-integer rounding gives the endpoints probability one eighth and each interior integer probability one fourth. The zero endpoint contributes nothing to the second moment.
E[Y2]=412+22+32+842=211
Compute
Compute
The rounded second moment is 5.5 and the continuous second moment is approximately 5.3333. Their absolute difference is one sixth.
