In a computer lab there are two di�erent printers. From 1pm-2pm Printer1 receives on average 12 print

jobs and Printer2 receives on average 5 print jobs. Assume the number of print jobs requested on Printer1
and on Printer2 are independent of each other.

(a) Defi�ne random variables to denote the number of print jobs each printer gets and state their distributions.

(b) What is the expected number of print jobs for Printer1 from 1:00pm-1:25pm?

(c) What is the probability that Printer2 receives 10 or more print jobs between 1pm and 2pm?

(d) What is the distribution of the number of print jobs requested in that computer lab from 1pm-2pm?

(e) What is the probability that there are 0 print jobs requested in the computer lab between 1pm and
2pm?

(a) To define the random variables for the number of print jobs each printer gets, we can use the following:

Let X be the random variable for the number of print jobs for Printer1. X follows a Poisson distribution with a mean of 12, as stated in the question.

Let Y be the random variable for the number of print jobs for Printer2. Y follows a Poisson distribution with a mean of 5, also stated in the question.

(b) To find the expected number of print jobs for Printer1 from 1:00 pm to 1:25 pm, we can calculate the expected value of the Poisson distribution. The expected value is equal to the mean of the distribution.

In this case, the mean of the Poisson distribution for Printer1 is 12. Therefore, the expected number of print jobs for Printer1 from 1:00 pm to 1:25 pm is 12.

(c) To find the probability that Printer2 receives 10 or more print jobs between 1 pm and 2 pm, we can calculate the cumulative probability of the Poisson distribution for Printer2.

P(Y ≥ 10) = 1 - P(Y < 10)

Using the Poisson probability formula, we can calculate P(Y < 10) for Printer2 with a mean of 5. Then subtract it from 1 to find P(Y ≥ 10).

(d) The distribution of the number of print jobs requested in the computer lab from 1 pm to 2 pm can be determined by considering the sum of the independent random variables X and Y.

If we let Z = X + Y, then Z represents the total number of print jobs requested in the computer lab. The distribution of Z will be the sum of the Poisson distributions for X and Y.

Therefore, the distribution of the number of print jobs requested in the computer lab from 1 pm to 2 pm is a convolution of two Poisson distributions.

(e) To find the probability that there are 0 print jobs requested in the computer lab between 1 pm and 2 pm, we need to calculate P(Z = 0), where Z represents the total number of print jobs in the lab as defined in part (d).

Using the convolution of the Poisson distributions for X and Y, we can calculate P(Z = 0). The formula for the probability mass function of the convolution of two Poisson distributions can be applied in this case.

P(Z = 0) = P(X = 0) * P(Y = 0)

Using the Poisson probability formula, we can calculate P(X = 0) and P(Y = 0) based on their respective means of 12 and 5. Then multiply these probabilities together to find P(Z = 0).