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Chapter 3: Q5E (page 98)
If the sample space S is an infinite set, does this necessarily imply that any rv X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.
No. Consider the random variable X defined as 鈥楪etting a Head鈥� in a random experiment of flipping a coin n times.
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Grasshoppers are distributed at random in a large field according to a Poisson process with parameter a \({\rm{\alpha = 2}}\) per square yard. How large should the radius \({\rm{R}}\) of a circular sampling region be taken so that the probability of finding at least one in the region equals \({\rm{.99}}\)?
A k-out-of-n system is one that will function if and only if at least k of the n individual components in the system function. If individual components function independently of one another, each with probability.\(9\), what is the probability that a 3-out-of-5 system functions?
A company that produces fine crystal knows from experience that 10% of its goblets have cosmetic flaws and must be classified as "seconds."
a. Among six randomly selected goblets, how likely is it that only one is a second?
b. Among six randomly selected goblets, what is the probability that at least two are seconds?
c. If goblets are examined one by one, what is the probability that at most five must be selected to find four that are not seconds?
Let X have a Poisson distribution with parameter \({\rm{\mu }}\). Show that E(X) =\({\rm{\mu }}\) directly from the definition of expected value. (Hint: The first term in the sum equals \({\rm{0}}\), and then x can be cancelled. Now factor out \({\rm{\mu }}\) and show that what is left sums to \({\rm{1}}\).)
Who studies more? Researchers asked the students in a large first-year college class how many minutes they studied on a typical weeknight. The back-to-back stemplot displays the responses from random samples of women and men from the class, rounded to the nearest minutes. Write a few sentences comparing the male and female distributions of study time.
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