ASSIGNMENT 2a Random Variables and Probability Distribution Function

1. Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords and the type of message, either email or voice. Also 70% of the messages arrive via email and the rest are voice.

number of keyword 0 1 2 3 4email 0.1 0.1 0.2 0.4 0.2

voice 0.3 0.4 0.2 0.1 0

Determine the probability mass function of the number of keywords in a message. [Ans: f(x0)= 0.16, f(x1)= 0.19, f(x2)= 0.2, f(x3)= 0.31, f(x4)= 0.14]

2. Determine the probability mass function for the random variable with the following cumulative distribution function:

F ( x )={0 x<20.20.50.81

2≤x<5.75.7≤x<6.56.5≤x<8.5x ≥8.5

[Ans: f(2)= 0.2, f(5.7)= 0.3, f(6.5)= 0.3, f(8.5)= 0.2]

3. From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that 25% of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful data? Assume that the company samples 5 customers.

[Ans: f(x0)= 0.2373, f(x1)= 0.3955, f(x2)= 0.26367, f(x3)= 0.08789, f(x4)= 0.014648, f(x5)= 0.000976]

4. The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly 9 of your calls are answered within 30

seconds? [Ans: 0.1877](b) If you call 20 times, what is the probability that at least 16 calls are answered in less than 30

seconds? [Ans: 0.4148](c) If you call 20 times, what is the mean number of calls that are answered in less than 30

seconds? [Ans: 15]

5. Traffic flow is traditionally modelled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of 6 cars per minute. To set the timing of a traffic signal the following probabilities are used.

(a) What is the probability of no cars through the intersection within 30 seconds? [Ans: 0.0498]

(b) What is the probability of three or more cars through the intersection within 30 seconds? [Ans: 0.5768]

(c) Calculate the minimum number of cars through the intersection so that the probability of this number or fewer cars in 30 seconds is at least 90%.[Ans: x=5]

(d) If the variance of the number of cars through the intersection per minute is 20, is the Poisson distribution appropriate? Explain. [Ans: not appropriate]