Independent solution

How to solve this Exponential Distribution question

Setup

Setup

For exponential scale beta, the pth quantile is obtained by setting its survival probability equal to one minus p.

Q(p)=−βln⁡(1−p),β=1,000,000Q(p)=-\beta\ln(1-p),\qquad \beta=1{,}000{,}000

Model

Model

Write the upper and lower quantiles and subtract them before rounding.

Q(0.95)=−βln⁡(0.05)Q(0.95)=-\beta\ln(0.05)
Q(0.15)=−βln⁡(0.85)Q(0.15)=-\beta\ln(0.85)

Compute

Compute

Combine the logarithms to evaluate the spread directly.

Q(0.95)−Q(0.15)=βln⁡(0.850.05)Q(0.95)-Q(0.15)=\beta\ln\left(\frac{0.85}{0.05}\right)
Q(0.95)−Q(0.15)=1,000,000ln⁡(17)=2,833,213.344Q(0.95)-Q(0.15)=1{,}000{,}000\ln(17)=2{,}833{,}213.344

Answer

Answer

Rounding the difference to the requested hundred-thousand unit gives 2,800,000.

2,800,000(D)\boxed{2{,}800{,}000\quad\text{(D)}}