need help with statistics homework

Posted on 2014-11-05 17:56:00 in Statistics by LINDA

1. In each scenario below, the discrete random variable X has either a geometric, binomial, negative binomial, or Poisson distribution. In each case, determine which of these four distributions describes the random variable X, and indicate the values of the parameters.
(a) Patients arrive in an emergency room at a rate of 4 per hour, and X is the number of patients who arrive during the next hour.
(b) A softball player gets a hit 26 percent of the time she bats, and X is the number of times she needs to bat to get her first hit.
(c) You roll a die repeatedly, and X is the number of rolls that it takes to get a 6 three times.
(d) A soccer team scores an average of 1.2 goals per game, and X is the number of goals that it scores in the next two games.
(e) A stock goes up independently each day with probability 0.57, and X is the number of days during the next two weeks that the stock goes up.

3. Suppose that fatal accidents occur in a national park at the rate of 2.3 per year.
(a) What is the probability that there will be at most two fatal accidents in the park during the next year.
(b) What is the probability that within the next 25 years, there will be at least one year with 6 or more fatal accidents?

4. Suppose that the number of goals that a soccer team scores in a game has a Poisson distribution with mean 1.2, and scores of different games are independent. What is the probability that the team will score at least one goal in exactly 6 of its next 10 games?

5. An elementary school tests all 800 of its students for possible hearing problems. Suppose each student is independently found to have a hearing problem with probability 0.003.
(a) Calculate the probability that at least three students in the school will be found to have a hearing problem?
(b) Use the Poisson approximation to the binomial to estimate the probability that at least three students in the school will be found to have a hearing problem.

6. Suppose that 50 people in a group are each given lottery tickets, with instructions to select three of the numbers between 1 and 20. Use the Poisson approximation to estimate the probability that there will be at least two people in the group who select the same three numbers. You may assume that everyone independently selects their three numbers at random.

7. An auto insurance company charges $800 per year for an insurance policy. Suppose that the company will have to pay out $40,000 each year to 1 percent of policy holders who suffer a serious injury in an accident, $3, 000 to 10 percent of policy holders whose vehicle is damaged in an accident, and nothing to the other 89 percent of policy holders. What is the expected profit that the company will make per year each time it sells a policy?

8. If you roll a die twice, what is the expected value of the smaller of the two numbers that you roll?

9. Suppose X is a random variable such that P(X = 1) = 2/3 and P(X = 3) = 1/3. (a) Calculate E[X3].
(b) Calculate E[eX ].

10. A couple decides to continue having children until they have one boy. Find the expected number of children they will have, the expected number of boys they will have, and the expected number of girls they will have.

12. The Bombardier De Havilland Dash 8-100 airplane seats 37 passengers. Suppose that it costs the airline $17,000 to fly the plane. Suppose that each passenger who has a ticket independently shows up for the flight with probability 0.94. The airline sells tickets for $500. If more passengers show up for the flight than there are seats available, then the airline must accommodate the extra passengers on a later flight and provide them with travel vouchers, at a cost to the airline of $1, 800 per extra passenger.
(b) What is the expected profit that the airline will make if it sells 38 tickets? (c) What is the expected profit that the airline will make if it sells 39 tickets?

13. Suppose X is a random variable taking its values in the set of positive integers, and suppose there is a positive number C and a number a > 1 such that
P(X = k) = Ck-a For what values of a is E[X] < 8?

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