Concrete mathematics : a foundation for computer science

Exercises
Warmups
8 EXERCISES 413
1 What’s the probability of doubles in the probability distribution Pro,
of (8.3), when one die is fair and the other is loaded? What’s the probability
that S = 7 is rolled?
2 What’s the probability that the top and bottom cards of a randomly shuffled
deck are both aces? (All 52! permutations have probability l/52!.)
3 Stanford’s Concrete Math students were asked in 1979 to flip coins until
Why only ten they got heads twice in succession, and to report the number of flips
numbers? required. The answers were
The other students
either weren’t 3, 2, 3, 5, ‘IO, 2, 6, 6, 9, 2.
empiricists or
they were just too Princeton’s Co:ncrete Math students were asked in 1987 to do a similar
Aipped out.
thing, with the following results:
10, 2, 10, 7, 5, 2, 10, 6, 10, 2.
Estimate the mean and variance, based on (a) the Stanford sample;
(b) the Princeton sample.
4 Let H(z) = F(z)/G(z), where F(1) = G(1) = 1. Prove that
Mean(H) = Mean(F) -Mean(G),
Var(H) = Var(F) - Var(G) ,
in analogy with (8.38) and (8.3g), if the indicated derivatives exist at
z= 1.
5 Suppose Alice and Bill play the game (8.78) with a biased coin that comes
up heads with probability p. Is there a value of p for which the game
becomes fair?
6 What does the conditional variance law (8.105) reduce to, when X and Y
are independent random variables?
Basics
7 Show that if two dice are loaded with the same probability distribution,
the probability of doubles is always at least i.
8 Let A and B be events such that A U B = f2. Prove that
Pr(wEAClB) = Pr(wEA)Pr(wEB)-Pr(w$A)Pr(w$B).
9 Prove or disprove: If X and Y are independent random variables, then so
are F(X) and G(Y), when F and G are any functions.