Independent solution

How to solve this Truncated Exponential Mean question

Setup

Setup

The mean of 5 gives rate 0.2, and the conditioning event has probability 1-exp(−2).

λ=0.2\lambda=0.2
Pr⁡(T<10)=1−e−2\Pr(T<10)=1-e^{-2}

Model

Model

Write the conditional first moment as a truncated integral divided by the conditioning probability.

E[T∣T<10]=∫010t(0.2)e−0.2t dt1−e−2E[T\mid T<10]=\frac{\int_0^{10}t(0.2)e^{-0.2t}\,dt}{1-e^{-2}}

Compute

Compute

Integration by parts gives a compact truncated-exponential mean formula.

E[T∣T<10]=5−10e−21−e−2E[T\mid T<10]=5-\frac{10e^{-2}}{1-e^{-2}}
=3.4348235725=3.4348235725

Answer

Answer

The conditional mean rounds to 3.435 years.

3.435(E)\boxed{3.435\quad\text{(E)}}