Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Subtract the lower endpoint to expose an ordinary exponential variable.

Y=X−dY=X-d
fY(y)=1βe−y/β,y≥0f_Y(y)=\frac1\beta e^{-y/\beta},\qquad y\ge0

Model

Model

The exponential density decreases from its lower endpoint, so the mode of X is d. Write the CDF of the shifted variable for the percentile calculation.

mode⁡(X)=d\operatorname{mode}(X)=d
FX(x)=1−e−(x−d)/β,x≥dF_X(x)=1-e^{-(x-d)/\beta},\qquad x\ge d

Compute

Compute

Set the CDF equal to 0.10 and solve for the percentile p.

0.10=1−e−(p−d)/β0.10=1-e^{-(p-d)/\beta}
e−(p−d)/β=0.90e^{-(p-d)/\beta}=0.90
p−d=−βln⁡(0.90)=βln⁡(109)p-d=-\beta\ln(0.90)=\beta\ln\left(\frac{10}{9}\right)

Answer

Answer

The lower endpoint cancels when the mode is subtracted from the percentile.

βln⁡(109)(B)\boxed{\beta\ln\left(\frac{10}{9}\right)\quad\text{(B)}}