Answer to Question #219944 in Statistics and Probability for reitu

Question

QUESTION 4

The time between customers arriving at a book store is exponentially distributed with a mean of 5

minutes.

(a) What is the probability that no customers arrive during a 30-minute period? (5)

(b) If the store opens at 9:00, what is the probability that the first customer of the day arrives

between 9:10 and 9:20? (6)

(c) If the store opens at 9:00, what is the probability that the first customer arrives at exactly

9:15? (4)

Answer 1

Suppose a random variable X denotes the time(in minutes) between customer's arrivals.

Therefore:

$X\approx Exp\left(\lambda =\frac{1}{5}=0.2\right)$

$f\left(x\right)=\lambda e^{-\lambda x}\:$

$f\left(x\right)\:=0.2e^{-0.2x}\:for\:x\ge 0$

(a). No customers arrive during a 30 - minute period if the the time between customer's arrivals is greater than 30 minutes.

Required probability is given by;

$P\left(X>30\right)=\int _{30}^{\infty }$ $0.2e^{-0.2x}dx\:$

$=\left[-e^{-0.2x}\:\right]^{^{\infty }}_{30}$

$=\left[e^{-0.2x}\:\right]^{^{30\:}}_{\infty }$

$=e^{-0.2\times 30}-e^{-\infty }$

$=e^{-6}-0\:$

$=0.0025$

Hence, $0.0025$ is the probability that no customers arrive during a 30 - minute period.

(b). Considering 9:00 as the origin of time(i.e. time zero), we obtain 9:10 as 10 minutes and 9:20 as 20 minutes . Required probability is given by;

$P\left(10<X<20\right)=\int _{10}^{20}0.2e^{-0.2x}\:dx$

$=\left[-e^{-0.2x}\right]^{^{20}}_{10}$

$=\left[e^{-0.2x}\right]^{^{10}}_{20}$

$=e^{-0.2\times 10}-e^{-0.2\times 20}$

$=e^{-2}-e^{-4}\:$

$=0.1353353-0.0183156$

$=0.1170$

Hence, $0.1170$ is the probability that the first customer arrives between 9:10 and 9:20.

(c). Considering 9:00 as the origin of time (i.e. time zero), we obtain 9:15 as 15 minutes.

Required probability is given by;

$P\left(X=15\right)=\int _{15}^{15}0.2e^{-0.2x}dx$

$=\left[e^{-0.2x}\right]^{^{15}}_{15}$

$=e^{-0.2\times 15}-e^{-0.2\times 25}$

$=e^{-3}-e^{-3}$

$=0.04978707-0.04978707$

$=0$

Hence, is the probability that the first customer arrives at exactly 9:15