Page 513 - Discrete Mathematics and Its Applications

492 7 / Discrete Probability

Exercises

1. What is the expected number of heads that come up when 17. Estimate the expected number of integers with 1000 dig-
a fair coin is flipped five times? its that need to be selected at random to find a prime,
2. What is the expected number of heads that come up when if the probability a number with 1000 digits is prime is
a fair coin is flipped 10 times? approximately 1/2302.
3. What is the expected number of times a 6 appears when 18. Suppose that X and Y are random variables and that
a fair die is rolled 10 times? X and Y are nonnegative for all points in a sample
4. A coin is biased so that the probability a head comes up space S. Let Z be the random variable defined by
when it is flipped is 0.6. What is the expected number of Z(s) = max(X(s), Y(s)) for all elements s ∈ S. Show
heads that come up when it is flipped 10 times? that E(Z) ≤ E(X) + E(Y).
5. What is the expected sum of the numbers that appear on 19. Let X be the number appearing on the first die when two
two dice, each biased so that a 3 comes up twice as often fair dice are rolled and let Y be the sum of the numbers ap-
as each other number? pearingonthetwodice.ShowthatE(X)E(Y) = E(XY).
6. What is the expected value when a $1 lottery ticket is ∗ 20. Show that if X 1 ,X 2 ,...,X n are mutually independent
n n
bought in which the purchaser wins exactly $10 million random variables, then E( i=1 X i ) = i=1 E(X i ).
iftheticketcontainsthesixwinningnumberschosenfrom The conditional expectation of the random variable X
the set {1, 2, 3,..., 50} and the purchaser wins nothing given the event A from the sample space S is E(X|A) =

otherwise? r · P(X = r|A).
r∈X(S)
7. The final exam of a discrete mathematics course con- 21. What is expected value of the sum of the numbers ap-
sists of 50 true/false questions, each worth two points, pearing on two fair dice when they are rolled given that
and 25 multiple-choice questions, each worth four points. the sum of these numbers is at least nine. That is, what
The probability that Linda answers a true/false question is E(X|A) where X is the sum of the numbers appearing
correctly is 0.9, and the probability that she answers a on the two dice and A is the event that X ≥ 9?
multiple-choice question correctly is 0.8. What is her ex- The law of total expectation states that if the sample
pected score on the final?
space S is the disjoint union of the events S 1 ,S 2 ,...,S n
8. Whatistheexpectedsumofthenumbersthatappearwhen and X is a random variable, then E(X) =
three fair dice are rolled? n E(X|S j )P(S j ).
j=1
9. Suppose that the probability that x is in a list of n distinct 22. Prove the law of total expectations.
integers is 2/3 and that it is equally likely that x equals 23. Use the law of total expectation to find the average weight
any element in the list. Find the average number of com- of a breeding elephant seal, given that 12% of the breed-
parisons used by the linear search algorithm to find x or ing elephant seals are male and the rest are female, and
to determine that it is not in the list.
the expected weights of a breeding elephant seal is 4,200
10. Suppose that we flip a fair coin until either it comes up pounds for a male and 1,100 pounds for a female.
tails twice or we have flipped it six times. What is the 24. Let A be an event. Then I A , the indicator random
expected number of times we flip the coin?
variable of A, equals 1 if A occurs and equals 0 oth-
11. Suppose that we roll a fair die until a 6 comes up or we erwise. Show that the expectation of the indicator ran-
have rolled it 10 times. What is the expected number of dom variable of A equals the probability of A, that is,
times we roll the die?
E(I A ) = p(A).
12. Suppose that we roll a fair die until a 6 comes up.
25. A run is a maximal sequence of successes in a se-
a) What is the probability that we roll the die n times? quence of Bernoulli trials. For example, in the sequence
b) What is the expected number of times we roll the die? S, S, S, F, S, S, F, F, S, where S represents success and
13. Suppose that we roll a pair of fair dice until the sum of F represents failure, there are three runs consisting of
the numbers on the dice is seven. What is the expected three successes, two successes, and one success, respec-
number of times we roll the dice? tively. Let R denote the random variable on the set of se-
14. Show that the sum of the probabilities of a random vari- quences of n independent Bernoulli trials that counts the
able with geometric distribution with parameter p, where number of runs in this sequence. Find E(R).[Hint: Show
n
I
0 <p ≤ 1, equals 1. that R = j=1 j , where I j = 1 if a run begins at the jth
15. Show that if the random variable X has the geometric Bernoulli trial and I j = 0 otherwise. Find E(I 1 ) and then
distribution with parameter p, and j is a positive integer, find E(I j ), where 1 <j ≤ n.]
then p(X ≥ j) = (1 − p) j−1 . 26. Let X(s) be a random variable, where X(s) is a nonneg-
16. Let X and Y be the random variables that count the num- ative integer for all s ∈ S, and let A k be the event that
∞
ber of heads and the number of tails that come up when X(s) ≥ k. Show that E(X) = k=1 p(A k ).
two fair coins are flipped. Show that X and Y are not 27. What is the variance of the number of heads that come up
independent. when a fair coin is flipped 10 times?