The continuous uniform standard deviation is 10/sqrt(12)=2.886751. The discrete uniform distribution on eleven consecutive integers has variance 10 and standard deviation 3.162278. Their absolute difference is 0.275526, which rounds to choice C.
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The continuous uniform standard deviation is 10/sqrt(12)=2.886751. The discrete uniform distribution on eleven consecutive integers has variance 10 and standard deviation 3.162278. Their absolute difference is 0.275526, which rounds to choice C.
Setup
Setup
Calculate the variance of the continuous uniform selection from its interval width.
Var(C)=12(10−0)2=325
SD(C)=1210=2.8867513459…
Model
Model
For the eleven equally likely integers, use the first two discrete moments.
E[D]=110+1+⋯+10=5
E[D2]=1102+12+⋯+102=35
Compute
Compute
Convert the discrete moments to a standard deviation and compare the two spreads.
Var(D)=35−52=10
SD(D)=10=3.1622776602…
∣SD(D)−SD(C)∣=0.2755263142…
Answer
Answer
The absolute difference rounds to 0.28.
0.28(C)
Error diagnosis
Why the other letters tempt
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis assumes equal endpoints and equal means force the continuous and discrete uniform distributions to have the same spread.
BThis effectively uses only ten discrete values by dropping one endpoint; the resulting standard-deviation difference is about 0.02.
DThis applies a half-unit continuity adjustment directly to a standard deviation instead of recomputing the discrete variance.
EThis subtracts the variances, 10-25/3=1.6667, rather than subtracting the standard deviations.
Original practice · fully worked
Original variant: hybrid calibration offset
A calibration offset X is generated in one of two ways. With probability 0.60 it is continuous uniform on [-2,2]. Otherwise it is selected uniformly from the three values -3, 0, and 3. Calculate the unconditional standard deviation of X.
A 1.155
B 1.673
C 1.789
D 2.449
E 3.200
Variant answer in brief
Both mixture components have mean zero. Their second moments are 4/3 and 6, so the unconditional variance is 0.60(4/3)+0.40(6)=3.2. Its square root is 1.788854, which rounds to choice C.
Setup
Setup
Both component distributions are symmetric about zero, so the mixture mean is zero.
E[X∣M=1]=E[X∣M=2]=0
E[X]=0
Model
Model
Calculate the conditional second moments for the continuous and discrete components.
E[X2∣M=1]=12(2−(−2))2=34
E[X2∣M=2]=3(−3)2+02+32=6
Compute
Compute
Average the component second moments, which equal their variances because both conditional means are zero.
Var(X)=0.60(34)+0.40(6)=3.2
SD(X)=3.2=1.7888543820…
Answer
Answer
The unconditional standard deviation rounds to 1.789.
1.789(C)
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