Concrete mathematics : a foundation for computer science

414 DISCRETE PROBABILITY
10 What’s the maximum number of elements that can be medians of a random
variable X, according to definition (8.7)?
11 Construct a random variable that has finite mean and infinite variance.
12 a If P(z) is the pgf for the random variable X, prove that
Pr(X $ r) < x.~‘P(x) for 0 < x < 1;
Pr(X 3 r) 6 x. -‘P(x) for x 3 1.
(These important relations are called the tail inequalities.)
b In the special case P(z) = (1 +~)“/2~, use the first tail inequality to
prove that t k,,,(z) 6 l/xan(l - CX)~'-~)~ when 0 < OL< i.
13 IfX,, . ..) Xln are inde:pendent random variables with the same distribution,
and if (x is any real number whatsoever, prove that
pr
(1
x1+...+xzn
2n
o1 < X1+-'fX,-K
3 1
IL1 n 1) 2'
14 Let F(z) and G(z) be probability generating functions, and let
H(z) = pF(z) + q G(z)
where p + q = 1. (This is called a miztzlre of F and G; it corresponds to
flipping a coin and choosing probability distribution F or G depending on
whether the coin comes up heads or tails.) Find the mean and variance
of H in terms of p, q, and the mean and variance of F and G.
15 If F(z) and G(z) are probability generating functions, we can define another
pgf H(z) by “composition”:
H(z) = F(G(z)).
Express Mean(H) and Var(H) in terms of Mean(F), Var(F), Mean(G),
and Var(G). (Equation (8.92) is a special case.)
16 Find a closed form for the super generating function En20 Fn(z)wn,
when F,(z) is the football-fixation generating function defined in (8.53).
17 Let X,,, and Yn,p have the binomial and negative binomial distributions,
respectively, with parameters (n, p). (These distributions are defined in
(8.57) and (8.60).) Prove that Pr(Y,,,