Problems using Probability Distribution of Random Variables

3-44
A real estate agent has four houses to sell before the end of the month by contacting prospective customers one by one. Each costumer has an independent 0.24 probability of buying a house on being contacted by the agent.

a) If the agent has enough time to contact only 15 customers, how confident can she be of selling all four houses within the available time?
b) If the agent wants to be a at least 70% confident of selling all the houses within the available time , at least how many customer should she contact?(if necessary, extend the template downward to more rows)
c) What minimum value of p will yield 70% confidence of selling all four houses by contacting at most 15 costumers?

3-52
Laptop computers produced by a company have an average life of 38.36 months. Assume that the life of a computer is exponentially distributed (which is a good assumption)
a) What is the probability that a computer will fail within 12 months?
b) If the company gives a warranty period of 12 months, what proportion of computers will fail during the warranty period?
c) Based on the answer (b), would you say the company can afford to give a warranty period of 12 months?
d) If the company wants not more than 5% of the computers to fail during the warranty period, what should be the warranty period?
e) If the company wants to give a warranty period of three months and stills wants not more than 5% of the computers to fail during the warranty period, what should be the minimum average life of the computers?

3-69
The number of orders for installation of a computer information system arriving at an agency per week is a random variable X with the following probability.
X P(x)
0 0.10
1 0.20
2 0.30
3 0.15
4 0.15
5 0.05
6 0.05
a) Prove that P (X) is a probability distribution
b) Find the cumulative distribution function of X
c) Use the cumulative distribution function to find probabilities P (2 < X &#8804; 5) , P (3 &#8804; X &#8804; 6) and P ( X> 4).
d) What is the probability that either four or five orders will arrive in a given week?
e) Assuming independence of weekly orders, what is the probability that three orders will arrive next week and the same number of orders the following week?
f) Find the mean and the standard deviation of the number of weekly orders.

3-77
Suppose that 5 of a total of 20 company accounts are in error. An auditor selects a random sample of 5 out of the 20 accounts. Let X be the number of accounts in the sample that are in error. Is X binomial? If not, what distribution does it have? If not, what distribution does it have? Explain.