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Chapter 4: Q.4.56 (page 167)
How many people are needed so that the probability that at least one of them has the same birthday as you is greater than ?
The probability the same birthday found to be.
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If X has distribution function F, what is the distribution function of the random variable 伪X + 尾, where 伪 and 尾 are constants,
An urn has n white and m black balls. Balls are randomly withdrawn, without replacement, until a total of white balls have been withdrawn. The random variable equal to the total number of balls that are withdrawn is said to be a negative hypergeometric random variable.
(a) Explain how such a random variable differs from a negative binomial random variable.
(b) Find .
Hint for (b): In order for to happen, what must be the results of the first withdrawals?
From a set of n elements, a nonempty subset is chosen at random in the sense that all of the nonempty subsets are equally likely to be selected. Let X denote the number of elements in the chosen subset. Using the identities given in Theoretical Exercise of Chapter, show that
Show also that for n large,
in the sense that the ratio Var(X) ton/approaches as n approaches q. Compare this formula with the limiting form of Var(Y) when P{Y =i}=/n,i=,...,n.
There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1 鈭� p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p, Ci, Ri, i = 1, 2, should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?
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