Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let X be exponential with rate 0.004 and let the reported benefit be the smaller of X and 250. First determine whether the uncapped median reaches the cap.

X∼Exp⁡(0.004)X\sim\operatorname{Exp}(0.004)
B=min⁡(X,250)B=\min(X,250)

Model

Model

Below 250 the capped and uncapped cumulative probabilities coincide, so solve the ordinary exponential median equation and then verify the resulting value is below 250.

P(B≤m)=P(X≤m)=1−e−0.004mfor m<250P(B\le m)=P(X\le m)=1-e^{-0.004m}\quad\text{for }m<250

Compute

Compute

Setting the cumulative probability to one half gives 173.2868. Because this is less than 250, capping does not alter the median.

1−e−0.004m=0.51-e^{-0.004m}=0.5
m=−log⁡(0.5)0.004=173.2868<250m=-\frac{\log(0.5)}{0.004}=173.2868<250

Answer

Answer

The median capped benefit rounds to 173, which is choice C.

173(C)\boxed{173\quad\text{(C)}}