Independent solution

How to solve this Exponential Distribution question

Answer in brief

Write both percentile conditions with the exponential CDF and eliminate the percentile level. The two survival values sum to one, giving y=-10 log(1-exp(-x/10)), which is choice A.

Setup

Setup

Use the exponential distribution function with mean 10 and rate 1/10.

F(t)=1−e−t/10F(t)=1-e^{-t/10}

Model

Model

Translate the two complementary percentile levels into distribution equations.

r=1−e−x/10r=1-e^{-x/10}
1−r=1−e−y/101-r=1-e^{-y/10}

Compute

Compute

Eliminate r, isolate the exponential term containing y, and take logarithms.

e−x/10+e−y/10=1e^{-x/10}+e^{-y/10}=1
e−y/10=1−e−x/10e^{-y/10}=1-e^{-x/10}
y=−10log⁡(1−e−x/10)y=-10\log\left(1-e^{-x/10}\right)

Answer

Answer

The expression has the mean 10 as the outer scale and the reciprocal mean in the exponent.

y=−10log⁡(1−e−x/10)(A)\boxed{y=-10\log\left(1-e^{-x/10}\right)\quad\text{(A)}}