Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let S(t) be the exponential survival probability. The reported four-hour median supplies the calibration S(4)=0.5.

S(t)=e−t/θS(t)=e^{-t/\theta}
S(4)=0.5S(4)=0.5

Model

Model

Exponential survival scales multiplicatively with time, so the five-hour survival is the four-hour survival raised to the ratio 5/4.

e−4/θ=0.5e^{-4/\theta}=0.5
S(5)=e−5/θ=(e−4/θ)5/4S(5)=e^{-5/\theta}=(e^{-4/\theta})^{5/4}

Compute

Compute

Raising 0.5 to the power 5/4 gives S(5)=0.420448.

S(5)=0.55/4=0.420448S(5)=0.5^{5/4}=0.420448

Answer

Answer

The five-hour survival probability rounds to 0.42, corresponding to choice D.

0.42(D)\boxed{0.42\quad\text{(D)}}