Answer
the average time between patient arrivals = $8$
probability of 2 patients arriving within 3 minutes of each other $\approx$ $0.31271$
Work Step by Step
the average time between patient arrivals
$\int_{0}^{\infty}tp(t)dt$ = $\int_{0}^{\infty}0.125te^{-0.125t}dt$ = $te^{-0.125t}|_{0}^{\infty}+\int_{0}^{\infty}e^{-0.125t}dt$ = $8$
probability of 2 patients arriving within 3 minutes of each other
$\int_{0}^{3}tp(t)dt$ = $\int_{0}^{3}0.125te^{-0.125t}dt$ =$-e^{-0.125t}|_{0}^{3}$ $\approx$ $0.31271$
