Question

A doctor has scheduled an appointment for patient A at 2:00 p.m. and an appointment for patient B at 2:40 p.m. The amounts of time that the appointments last are independent, identically distributed, exponential random variables with mean 40 minutes. Assuming that both patients arrive exactly on time,

(a) Calculate the probability that patient A will still be in the doctor’s office at 2:45 p.m. (b) Find the expected amount of time that patient B spends at the doctor’s office.

Answer 1

Let X denote the time that an appointment lasts (in minutes). Then

X ~exp(X=

So, $F(x)=1-e^{-\lambda x} =1-e^{-\frac{x}{40}},x>0$

a)

Required probability = P(appointment last for more than 45 minutes)

$=P(X>45)=1-P(X<45)= 1-[1-e^{-\frac{45}{40}}]=0.3247$

b)

We know that

X ~exp(X=

So, Expected amount of time that patient B spends at the doctor’s office = $E(X)=\frac{1}{\lambda }=40$

i.e. 40 minutes