The following examples are experiments and their associated random variables. In each case identify the values the random variable can assume and state whether the random variable is discrete or continuous:
Problems 91
Experiment Random Variable (x)
a. Take a 20-question examination Number of questions answered correctly
b. Observe cars arriving at a tollbooth Number of cars arriving at the tollbooth for 1 hour
c. Audit 50 tax returns Number of returns containing errors
d. Observe an employee's work Number of nonproductive hours for 8 hours
e. Weigh a shipment of goods Number of pounds
3. The following examples are experiments and their associated random variables. In each case identify the values the random variable can assume and state whether the random variable is discrete or continuous:
Problems 91
Experiment Random Variable (x)
a. Take a 20-question examination Number of questions answered correctly
b. Observe cars arriving at a tollbooth Number of cars arriving at the tollbooth for 1 hour
c. Audit 50 tax returns Number of returns containing errors
d. Observe an employee's work Number of nonproductive hours for 8 hours
e. Weigh a shipment of goods Number of pounds
8. The J. R. Ryland Computer Company is considering a plant expansion that will enable the company to begin production of a new computer product. The company's president must determine whether to make the expansion a medium- or large-scale project. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.50, and 0.30, respectively. Letting x indicate the annual profit in $1000s, the firm's planners developed profit forecasts for the medium- and large-scale expansion projects.
Medium-Scale Large-Scale
Expansion Profits Expansion Profits
x f(x) y f(y)
Low 50 0.20 0 0.20
Demand Medium 150 0.50 100 0.50
High 200 0.30 300 0.30
a. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit?
b. Compute the variance for the profit associated with the two expansion alternatives.
Which decision is preferred for the objective of minimizing the risk or uncertainty?
9. Consider a binomial experiment with 2 trials and p =0.4.
a. Compute the probability of 1 success, f(1).
b. Compute f(0).
c. Compute f(2).
d. Find the probability of at least one success.
e. Find the expected value, variance, and standard deviation
14. Consider a Poisson probability distribution with 2 as the average number of occurrences per time period.
a. Write the appropriate Poisson probability function.
b. What is the average number of occurrences in three time periods?
c. Write the appropriate Poisson probability function to determine the probability of occurrences in three time periods.
d. Find the probability of two occurrences in one time period.
e. Find the probability of six occurrences in three time periods.
f. Find the probability of five occurrences in two time periods.
18. Consider a Poisson probability distribution with 2 as the average number of occurrences per time period.
a. Write the appropriate Poisson probability function.
b. What is the average number of occurrences in three time periods?
c. Write the appropriate Poisson probability function to determine the probability of occurrences in three time periods.
d. Find the probability of two occurrences in one time period.
e. Find the probability of six occurrences in three time periods.
f. Find the probability of five occurrences in two time periods.
21. For the standard normal random variable z, compute the following probabilities:
a. P(0 <- z <- 0.83)
b. P(-1.57 <- z <-0)
c. P(z > 0.44)
d. P(z >-20.23)
e. P(z < 1.20)
f. P(z <-20.71)
29. Consider the exponential probability density function:
http://goo.gl/BBFGCH
a. Write the formula for P(x <-x0).
b. Find P(x <- 2).
c. Find P(x >- 3).
d. Find P(x <- 5).
e. Find P(2 <-x <-5).
ch 4.
1. The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature:
Problems 135
State of Nature
Decision Alternative s1 s2 s3
d1 250 100 25
d2 100 100 75
State of Nature
Decision Alternative s1 s2 s3 s4
d1 14 9 10 5
d2 11 10 8 7
d3 9 10 10 11
d4 8 10 11
13a. Construct a decision tree for this problem.
b. If the decision maker knows nothing about the probabilities of the three states of nature, what is the recommended decision using the optimistic, conservative, and minimax regret approaches?
5.The following profit payoff table was presented in Problem 1. Suppose that the decision
maker obtained the probability assessments P(s1) = 0.65, P(s2) = 0.15, and P(s3) =0.20.
Use the expected value approach to determine the optimal decision.
Problems 137
State of Nature
Decision Alternative s1 s2 s3
d1 250 100 25
d2 100 100 75