Question

4. * Suppose that jobs are sent to a printer at an average rate of 10 per hour. (a) Let X = the number of jobs sent in an hour. What is the distribution of X? Give the name and parameter values. (b) What is the probability that exactly 8 jobs are sent to the printer in an hour? (c) Let X = the number of jobs sent in a 12 min period. What is the distribution of X? Give the name and parameter values. (d) What is the probability that at least 3 jobs will be sent to the printer in a 12 min period? (e) How many jobs do you expect to be sent to the printer in a 12 min period?

Answer 1

(a)

The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period

if average number of times the event occurs over that time period then probability mass function:

Expected value of Poisson i.e

E(X) =

X : number of jobs sent in an hour

jobs are sent to a printer at an average rate of 10 per hour i.e

: Average number of jobs sent in an hour = 10

Therefore

Distribution of X is Poisson.

Parameter = 10

Probability mass function of X =

'e' value approximately 2.71828

(b) Probability that exactly 8 jobs are sent to the printer in an hour = P(X=8)=P(8)

Probability that exactly 8 jobs are sent to the printer in an hour = 0.1126

(c)

X = the number of jobs sent in a 12 min period

Given,

Average number of jobs sent in an hour = 10

i.e

Average number of jobs sent in a 12 min period

: Average number of jobs sent in a 12 minute period= 2

Distribution of X is Poisson.

Parameter = 2

Probability mass function of X

(d) probability that at least 3 jobs will be sent to the printer in a 12 min period = P(X3)

P(X3) = 1- P(X<3)

P(X<3) = P(X=0)+P(X=1)+P(X=2)

P(X<3) = P(X=0)+P(X=1)+P(X=2) = 0.1353+0.2707+0.2707 = 0.6767

P(X3) = 1- P(X<3) = 1- 0.6767 = 0.3233

probability that at least 3 jobs will be sent to the printer in a 12 min period = 0.3233

(e) How many jobs do you expect to be sent to the printer in a 12 min period i.e

Expected value of X : E(X)

E(X) of Poisson distribution =

For the given problem, =2

Therefore, Expected value of X : E(X)=2

Number of jobs expected to sent to printer : E(X) = 2

Ans : 2