Original practice · fully worked
Original variant: distribution of a two-tier shipping rebate
Shipping cost X is exponential with mean 50 dollars. A rebate is zero through 10 dollars, pays all cost from 10 through 40, and then pays half of additional cost. Given a positive rebate, find the probability the rebate is below 45 dollars.
- D
0.6988 - B
0.7534 - C
0.7981 - A
0.8347 - E
0.8647
Answer and calculation
A 45-dollar rebate corresponds to cost 70. Conditional on X>10, the probability X<70 is 1-exp(-(70-10)/50)=0.6988.
Setup
Setup
A positive rebate corresponds to shipping cost above 10 dollars. Inverting the two-tier schedule shows that a rebate below 45 corresponds to cost below 70.
Y>0⟺X>10 Y<45⟺X<70 Model
Model
Condition on cost above 10 and use exponential memorylessness over the 60-dollar interval from 10 to 70.
P(Y<45∣Y>0)=P(X<70∣X>10) Compute
Compute
The conditional cumulative probability is 1 minus exp(-60/50), equal to 0.698806.
P=1−e−(70−10)/50=1−e−1.2=0.698806 Answer
Answer
Thus the conditional probability is 0.6988, corresponding to choice D.
0.6988(D) 