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Chapter 7: Q.7.1 (page 359)
Show that is minimized at .
Differentiaterespective to.
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If are independent and identically distributed random variables having uniform distributions over , find
(a) ;
(b) .
A fair die is rolled times. Calculate the expected sum of the rolls.
7.4. If X and Y have joint density function find
(a) E[X Y]
(b) E[X]
(c) E[Y]
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.
Cards from an ordinary deck of playing cards are turned face upon at a time. If the 1st card is an ace, or the nd a deuce, or the rd a three, or ...,or the th a king,or the an ace, and so on, we say that a match occurs. Note that we do not require that the (n + ) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.
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