Independent solution

How to solve this Exponential Distribution question

Setup

Setup

The lifetime is exponential with mean 300, and the target is its conditional 95th percentile after survival beyond 100.

X∼Exp⁡(mean 300)X\sim\operatorname{Exp}(\text{mean }300)
P(X≤p∣X>100)=0.95P(X\le p\mid X>100)=0.95

Model

Model

By memorylessness, the excess beyond 100 is exponential with the same mean. A 95th conditional percentile leaves upper-tail probability 0.05.

P(X>p∣X>100)=e−(p−100)/300=0.05P(X>p\mid X>100)=e^{-(p-100)/300}=0.05

Compute

Compute

Inverting the 0.05 tail and restoring the elapsed 100 gives 998.719.

p=100−300log⁡(0.05)=998.719p=100-300\log(0.05)=998.719

Answer

Answer

The requested conditional percentile rounds to 1000, corresponding to choice E.

1000(E)\boxed{1000\quad\text{(E)}}