Question

Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 1/ 8. If someone arrives immediately ahead of you at a public telephone booth, find the probability that
(a) you will have to wait more than 5 minutes.

(b) you will have to wait between 10 and 20 minutes.

(c) If you have waited for 5 minutes, what is the probability that you will have to wait more than 15 minutes in total?

Answer 1

Answer:

a) P=0.535

b) P=0.204

c) P=0.286

Step-by-step explanation:

The exponential distribution is expressed as

$F(x>t)=e^{-\lambda t}$

In this example, λ=1/8=0.125 min⁻¹.

a) The probability of having to wait more than 5 minutes

$F(x>5)=e^{-0.125*5}=e^{-0.625}=0.535$

b) The probability of having to wait between 10 and 20 minutes

$F(1020)=e^{-0.125*10}-e^{-0.125*20}\\\\F(10$

c) The exponential distribution is memory-less, so it is independent of past events.

If you have waited 5 minutes, the probability of waiting more than 15 minutes in total is the same as the probability of waiting 15-5=10 minutes.

$F(x>15|t^*=5)=F(x>15)/F(x>5)=\frac{e^{-0.125*15}}{e^{-0.125*5}}=e^{-0.125*(15-5)}\\\\ F(x>15|t^*=5)=e^{-0.125*(10)}=F(x>10)=0.286$