Question

A computer lab has two printers. Printer I handles 40% of all the jobs. Its printing time is Exponential with the mean of 2 minutes. Printer II handles the remaining 60% of jobs. Its printing time is Uniform between 0 minutes and 5 minutes. A job was printed in less than 1 minute. What is the probability that it was printed by Printer I?

Answer 1

Answer:  0.3042

Explanation:

Let A  and B are the events to that job done by Printer I and Printer II respectively.

Given : P(A)=0.40    P(B)=0.60

Printing time of Printer I is Exponential with the mean of 2 minutes.

i.e. average number of job done in one minute:$\lambda=\dfrac{1}{2}$

The cumulative distribution function (CDF) for exponential distribution:-

$F(x)=1-e^{-\lambda x}$, where $\lambda$ is the mean.

Then, the cumulative distribution function (CDF) for Printer I:-

$P(X|A)=1-e^{-\dfrac{1}{2} x}$

i.e. $P(X$

Printing time of Printer II is Uniform between 0 minutes and 5 minutes.

The cumulative distribution function  (CDF) for uniform distribution in interval (a,b) :-

$F(x)=\dfrac{x-a}{b-a}$

Then, $P(X|B)=\dfrac{x-0}{5-0}=\dfrac{x}{5}$

i.e. $P(X$

Now, the required probability :-

$\text{P(A}|X$

Hence, the required probability = 0.3042