Probability Statistic Problems

Sample Problems - Please show all your work.
QUESTION 1

Suppose that you are working for an energy company that distributes natural gas to residents in a city. As the chief economic analyst, you are asked to analyze the demand for natural gas in your city. Let X denote the random monthly demand for natural gas in millions of cubic feet. X takes on a value of either 10 or 20. The probability that the demand is 10 million cubic feet is 0.6.

a. Is X a continuous or discrete random variable? Why?

b. What is the expected value of the monthly demand?

c. Suppose that the demand for gas in a given month is independent of the demand in all other months. Calculate the probability that the demand is equal to 20 in 10 out of 12 months in a year.

d. Calculate the expected value of the total demand in a given year.

QUESTION 2
You are considering to invest in a portfolio which can have three possible yields with the
Following probabilities:

X $1,000 $500 $2,000
P (x) 0.5 0.4 0.1

a. Find the probability that the yield is more than $500.

b. Find the average yield of the portfolio.

c. Find the standard deviation of the portfolio.

QUESTION 3
The city of Houston is considering introducing a light rail system. Suppose that a city committee of n = 5 members vote whether to proceed with the construction of the rail system or not. The decision will be based on majority rule, i.e. if 3 or more members of the committee are decisive about one particular action, then that action will be taken. Assume that each member of the committee votes in favor of the proposal with probability p =0.5. Denote by X the total number of members who vote in favor of the proposal.

a. What kind of a random variable is X? Justify your answer.

b. What is the expected value of the random variable X?

c. What is the standard deviation of the random variable X?

d. What is the probability that the proposal will pass?

QUESTION 4
The average number of accidents that take place on a particularly dangerous 20-mile stretch of the highway I-10 during a period of one year is 5. Denote by X the random number of accidents observed.

a. What kind of random variable is X? Justify your answer.

b. Find the probability that there are 7 accidents.

c. Find the probability that there is at least one accident.

QUESTION 5
Assume that among families with two children there are equal numbers of families with (boy, boy), (boy, girl), (girl, boy), (girl, girl), where the ordering in the parentheses indicates the order of birth. We select a family with two children at random.

a. What is the probability that the family has 2 boys, given that it has at least one boy?

b. Suppose you are married and currently have one child, a boy. You are expecting a second child. Clearly, once you have two children, you will have at least one boy. Is the answer to part (a) the same as the probability that your second child will be a boy?

QUESTION 6
It is, of course, possible to model an experiment with independent trials, but with unequal
Probabilities of success from trial to trial. Assume n = 2 independent trials are to be performed; the probability of success for the first trial is .4 and the probability of success for the second is .8. Let X be the total number of successes observed and evaluate its probability distribution function, mean, and variance. Is X Binomial?

QUESTION 7
A radio-television dealer extends credit to people buying his sets. His past experience shows that 10 percent of all those to whom he extended credit did not pay and he took a loss on each such sale. Assume that the dealer has 5 identical TV sets and that he will sell them on credit to 5 people. If the buyer does not pay he takes a loss of $200. If the buyer pays in full he makes a profit of $100.

a. What is the probability distribution function of the total net profit, X, he will make on these 5 sales? (Hint: You have to provide the probabilities for each possible amount of profit)

b. What is the expected value of the total net profit?

c. What is the standard deviation of the total net profit?

QUESTION 8
Suppose that a printed page in a book contains 40 lines, and each line contains 75 positions (each of which may be blank or filled with a symbol). Thus each page contains 3000 positions to be set. Assume that a particular typesetter makes one error per 6000 positions on the average.

a. What is the distribution of X, the number of errors per page? Why?

b. Compute the probability that a page contains no errors.

c. What is the probability that a 16-page chapter contains no errors?

QUESTION 9
Suppose that 4 cities are bidding for 2012 Olympics: Houston (H), Dallas (D), Baltimore (B), and New York (N). The winner is selected in two stages. In the first stage, 3 cities from U.S. are selected. In the second stage, the selected U.S. cities compete with 5 other non-US cities to become the winner. Suppose that the selections in the first stage are independent of each other and each city is equally likely to be selected. Furthermore, assume that each city in the second stage is also equally likely to become the winner.

a. Find the probability that both Texas cities make it to the second stage.

b. Find the probability that New York does not make it to the second stage.

c. What is the probability that Houston becomes the site for Olympics?

d. Find the probability that a Texas City becomes the site for Olympics.

QUESTION 10
Suppose that Y is a binomial random variable with mean = 6 and variance = 4.

a. What is n and p for this binomial random variable?

b. Find the probability that Y <18.