This Exam P sample reference tests Mixed-Distribution Expectation. The CDF represents equal weight on a point mass at zero and a uniform distribution from 2 to 3. The mixture mean is 0.5(0)+0.5(2.5)=1.25, choice D.
How to solve this Mixed-Distribution Expectation question
Setup
Setup
The distribution is mixed: half of its probability is concentrated at zero and the remaining half is spread uniformly over the interval from two to three.
Pr(X=0)=0.5,fX(x)=0.5for 2<x<3
Model
Model
Compute the expectation by adding the point-mass contribution and the integral of the continuous component.
E[X]=0(0.5)+∫230.5xdx
Compute
Compute
The point mass contributes zero, while the continuous component contributes 1.25.
E[X]=0.5(232−22)=1.25
Answer
Answer
The expected value is 1.25.
1.25(D)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0 uses only the zero point mass and discards the continuous component.
BThe value 0.50 is the probability weight of the continuous component, not its contribution to the mean.
EThe value 2.50 is the conditional mean of the uniform component and ignores that it occurs with probability one half while the remaining mass is at zero.
Original practice · fully worked
Original variant: mean of a calibration mixture
A calibration error equals 1 with probability 0.30. With the remaining probability 0.70 it is uniformly distributed between 3 and 5. Calculate the expected calibration error.
A 2.4
B 2.8
C 3.1
D 3.4
E 4.0
Variant answer in brief
The point-mass contribution is 0.30 and the uniform component contributes 0.70(4)=2.80, giving mean 3.10, choice C.
Setup
Setup
Separate the point mass at one from the uniform component on the interval from three to five. The uniform component has mean four.
E[U]=23+5=4
Model
Model
The expectation of a mixture is the probability-weighted average of its component means.
E[X]=0.30(1)+0.70E[U]
Compute
Compute
The point mass contributes 0.30 and the uniform component contributes 2.80, giving a total mean of 3.10.
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