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Chapter 4: Q.4.9 (page 170)
Show how the derivation of the binomial probabilities leads to a proof of the binomial theorem when and are nonnegative.
Hint: Let .
Assume the hint and the point that the totality of all possible probabilities must equal one.
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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.
Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter 位 = 3.
(a) Find the probability that 3 or more accidents occur today.
(b) Repeat part (a) under the assumption that at least 1 accident occurs today.
Two fair dice are rolled. Let equal the product of the dice. Compute .
If X has distribution function F, what is the distribution function of the random variable 伪X + 尾, where 伪 and 尾 are constants,
A person tosses a fair coin until a tail appears for the first time. If the tail appears on the n th flip, the person wins 2n dollars. Let X denote the player's winnings. Show that . This problem is known as the St. Petersburg paradox.
(a) Would you be willing to pay \( 1 million to play this game once?
(b) Would you be willing to pay \) 1 million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?
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