Independent solution

How to solve this Exponential Distribution question

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)}}