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Chapter 5: Q 5.49. (page 209)
What does it mean three events to be mutually exclusive.?
.If three occurrences have no common consequences, they are said to be mutually exclusive.
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Discuss the pros and cons of binomial probability tables.
Expected Utility. One method for deciding among various investment involves the concept of expected utility. Economists describe the importance of various levels of wealth by using utility functions. For instance, in most case, a single dollar is more important (has greater utility ) for someone with little wealth than for someone with greater wealth Consider two investments, say investment A and B. Measured in thousand of dollars, suppose that investment A yields 0, 1, and 4 with probability 0.1 and16 with probability 0.5, 0.3 and 0.2 respectively. Let Y denote the yield of an investment. For the two investment, determine and compare.
Part (a) The mean of Y, the expected yield.
Part (b) The mean of ,the expected utility, using the utility function role="math" localid="1651845051902" Interpret the utility function
Part (c) The mean of ,the expected utility, using the utility function . Interpret the utility function v
A bowl contains 12 poker chips 3 red , 4 white and 5 blue. One of these poker chips is selected at random from the bowl. Let B denote the event that the chips is selected is blue. Find the probability that a blue chips is selected, and express your answer in probability notation
The Hypergeometric Distribution. In this exercise, we discuss the hypergeometric distribution in more detail. When sampling is done without replacement from a finite population, the hypergeometric distribution is the exact probability distribution for the number of members sampled that have a specified attribute. The hypergeometric probability formula is
,
where Xdenotes the number of members sampled that have the specified attribute, Nis the population size, nis the sample size, and pis the population proportion.
To illustrate, suppose that a customer purchases 4 fuses from a shipment of 250, of which 94 % are not defective. Let a success correspond to a fuse that is not defective.
(a) Determine N, n, and p.
(b) Apply the hypergeometric probability formula to determine the probability distribution of the number of nondefective fuses that the customer gets.
Key Fact 5.6 shows that a hypergeometric distribution can be approximated by a binomial distribution, provided the sample size does not exceed 5% of the population size. In particular, you can use the binomial probability formula
with , to approximate the probability distribution of the number of nondefective fuses that the customer gets.
(c) Obtain the binomial distribution with parameters .
(d) Compare the hypergeometric distribution that you obtained in part (b) with the binomial distribution that you obtained in part (c).
Fill in the blanks.
(a) A is a quantitative variable whose value depends on chance.
(b) A discrete random variable is a random variable whose possible values .
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