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Chapter 7: Q.28 (page 361)
Show that.
We prove that,
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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.
In Problem 7.6, calculate the variance of the sum of the rolls.
A prisoner is trapped in a cell containingdoors. The first door leads to a tunnel that returns him to his cell after days鈥� travel. The second leads to a tunnel that returns him to his cell after days鈥� travel. The third door leads to freedom after day of travel. If it is assumed that the prisoner will always select doors and with respective probabilities and ., what is the expected number of days until the prisoner reaches freedom?
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).
Suppose that each of the elements of is to be colored either red or blue. Show that if are subsets of , there is a way of doing the coloring so that at most of these subsets have all their elements the same color (where denotes the number of elements in the set ).
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