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Chapter 7: Q.52 (page 356)
A population is made up of disjoint subgroups. Let denote the proportion of the population that is in subgroup . If the average weight of the members of subgroup is , what is the average weight of the members of the population?
The average weight of the members of the population is
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Let be independent random variables having an unknown continuous distribution function and let be independent random variables having an unknown continuous distribution function . Now order those variables, and let
The random variable is the sum of the ranks of the sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether and are identical distributions. This test accepts the hypothesis that when is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of .
Hint: Use the results of Example 3e.
Prove Proposition when
and have a joint probability mass function;
and have a joint probability density function and
for all .
There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, .and .. One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the flips?
Suppose that the expected number of accidents per week at an industrial plant is . Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of . If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week .
We say that is stochastically larger than , written , if, for all ,
Show that if then when
(a) and are nonnegative random variables;
(b) and are arbitrary random variables. Hint:
Write as
where
Similarly, represent as . Then make use of part (a).
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