Original practice · fully worked
Original variant: variance of a capped discrete delay score
A delivery delay T equals one hour on 20% of trips, four hours on 50%, and seven hours on the other 30%. A dashboard records Y=min(T,5), so the longest delay is entered as five hours. Find Var(Y).
- A
1.21 - B
1.70 - C
2.01 - D
2.49 - E
3.30
Answer and calculation
The recorded values are 1, 4, and 5. Their mean is 3.7 and second moment 15.7, so the variance is 2.01.
Setup
Setup
Capping the seven-hour delay at five changes the three possible recorded values to 1, 4, and 5 while preserving their probabilities.
Y∈{1,4,5} with probabilities 0.2,0.5,0.3 Model
Model
Compute the first two moments directly from the three-point distribution of the recorded value Y.
E[Y]=0.2(1)+0.5(4)+0.3(5)=3.7 E[Y2]=0.2(1)+0.5(16)+0.3(25)=15.7 Compute
Compute
The first moment is 3.7 and the second raw moment is 15.7, so subtracting the squared mean gives variance 2.01.
Var(Y)=15.7−(3.7)2=2.01 Answer
Answer
Thus the dashboard-recorded delivery delay has variance 2.01, corresponding to choice C.
2.01(C) 