Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Identify the exponential scale from the density and use its equal mean and standard deviation.

X∼Exponential⁡(mean 20)X\sim\operatorname{Exponential}(\text{mean }20)
E[X]=20,SD⁡(X)=20E[X]=20,\qquad \operatorname{SD}(X)=20

Model

Model

Half a standard deviation is ten, so translate the centered-distance condition into endpoints on the positive support.

∣X−20∣<10⟺10<X<30\lvert X-20\rvert<10\quad\Longleftrightarrow\quad 10<X<30

Compute

Compute

Use the exponential survival function at the two endpoints.

Pr⁡(10<X<30)=Pr⁡(X>10)−Pr⁡(X>30)\Pr(10<X<30)=\Pr(X>10)-\Pr(X>30)
=e−10/20−e−30/20=e−1/2−e−3/2=0.3834004996=e^{-10/20}-e^{-30/20}=e^{-1/2}-e^{-3/2}=0.3834004996

Answer

Answer

The interval probability rounds to 0.38.

0.38(B)\boxed{0.38\quad\text{(B)}}