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Chapter 4: Q.4.37 (page 166)
Find Var(X) and Var(Y) for X and Y as given in Problem 4.21
In the given information the answers are
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A certain typing agency employs typists. The average number of errors per article is when typed by the first typist and when typed by the second. If your article is equally likely to be typed by either typist, approximate the probability that it will have no errors.
A satellite system consists of components and functions on any given day if at least of the n components function on that day. On a rainy day, each of the components independently functions with probability whereas, on a dry day, each independently functions with probability . If the probability of rain tomorrow is what is the probability that the satellite system will function?
In Problem for if the coin is assumed fair, what are the probabilities associated with the values that X can take on?
Let be a negative binomial random variable with parameters and , and let be a binomial random variable with parameters and . Show that
Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity
or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events to express the events and in terms of the outcomes of this sequence.
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.
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