53
5.4. PROBLEMS
4.2.1 Pick a point x at random (with uniform density) in the interval
[0, 1]. Find the probability that x > 1/2 gived that
(a) x > 1/4,
(b) x < 3/4,
(c ) | x 1 / 2 | < 1 / 4 ,
(d) x2 x + 2/9 < 0.
For any events A, B we have
P (A|B ) =
P (
Solutions to Problem Set #7
Section 8.1
1. A fair coin is tossed 100 times. The expected number of heads is 50, and the standard
deviation for the number of heads is (100 1/2 1/2)1/2 = 5. What does Chebyshevs inequality
tell you about the probability that
Solutions to Problem Set #8
Section 9.1
8. A club serves dinner to members only. They are seated at 12-seat tables. The manager
observes over a long period of time that 95 percent of the time there are between six and nine
full tables of members, and the
19
3.3. PROBLEMS
3.3
Problems
2.2.1 (a) Presumably the authors mean that the probability of landing in an
interval is proportional to the length of the interval. This means f (x) = C
for some positive constant C . Since
10
1 = P () =
C dx = 8C,
2
we must
Solutions to Problem Set #4
Section 4.1
18. A doctor assumes that a patient has one of three diseases d1 , d2 or d3 . Before any test, he
assumes an equal probability for each disease. He carries out a test that will be positive with
probability .8 if the
Solutions to Problem Set #6
Section 5.1
10.(Paraphrased) The US Census wants to estimate the number of people not counted by the
census. In a given locality, let N be actual the number of people who live there and let n1 be
the number of people counted by
Solutions to Problem Set #2
Section 3.1
3. In a digital computer, a bit is one of the integers cfw_0, 1, and a word is any string of 32
bits. How many dierent words are possible?
This problem is analogous to the menu problem from class. In this case, ther
Solutions to Problem Set #1
1
1. Let = cfw_a, b, c be a sample space. Let m(a) = 2 , m(b) =
probabilities for all eight subsets of .
1
3
1
and m(c) = 6 . Find the
The eight subsets are: , cfw_a, cfw_b, cfw_c, cfw_a, b, cfw_a, c, cfw_b, c, cfw_a, b, c.
m()
Solutions to Problem Set #2
Section 3.2
6. Charles claims he can distinguish between beer and ale 75% of the time. Ruth bets he
cannot. A bet is made: Charles is to be given ten small glasses, each having been lled with
beer or ale, chosen by tossing a fa
48
CHAPTER 5. CONDITIONAL PROBABILITY
5.4
Problems
4.1.1 Suppose E and F are events with positive probabilities. Show that if
P (E |F ) = P (E ), then P (F |E ) = P (F ).
Suppose
P (E F )
P (E | F ) =
= P (E ).
P (F )
Then
P (E F ) = P (F )P (E ).
Thus
P
78
7.2
CHAPTER 7. EXPECTED VALUE AND VARIANCE
Problems
6.1.2 If a red card is drawn the player wins $1. If a black card is drawn the
player loses $2. What is the expected value?
1
1
1
2 = .
2
2
2
6.1.4 In Las Vegas the roulette wheel has 0, 00, and the nu
93
7.5. PROBLEMS
7.5
Problems
6.3.2 Let X be a random variable with range [1, 1] and density function
fX . Find (X ) and 2 (X ) if fX (x) = 0 for |x| > 1, and for |x| 1
Since the interval [1, 1] is symmetric about x = 0, any even density
(f (x) = f (x) wi
86
7.3.5
CHAPTER 7. EXPECTED VALUE AND VARIANCE
Problems
6.2.2 A random variable X has the distribution
0
1
2
4
1 / 3 1/ 3 1/ 6 1/ 6
pX =
Find the expected value, variance, and standard deviation of X .
1
1
1
1
8
+1 +2 +4 = .
3
3
6
6
6
1
1
1
22
1
E (X 2 )
74
CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES
6.2
Problems
5.1.1 Which are modeled with a uniform distribution?
(a) Yes, P (X = k ) = 1/6 for k = 1, . . . , 6.
(b) No, this has a binomial distribution.
(c) Yes, P (X = k ) = 1/38 for k = 0, 00, 1, .
26
4.2
CHAPTER 4. COMBINATORICS
Problems
3.1.1 There are 4 3 2 1 = 4! = 24 ways to line up 4 people.
3.1.2 There are 4 3 = 12 ways to produce (exterior, interior ) color
combinations.
3.1.3 The number of length 32 bit strings is 2 2 2, with 32 stages
of t
Solutions to Problem Set #8
Section 11.2
6. In the Land of Oz example (also known as the weather in Hanover example from
class), change the transition matrix by making R an absorbing state. This gives
1
0
0
P = 1/2 0 1/2 .
1/4 1/4 1/2
Find the fundamental
Solutions to Problem Set #4
Section 6.1
14. In the hat check problem, N people check their hats and the hats are handed back at
random. Let Xj = 1 if the j th person gets his or her own hat and 0 otherwise. Find E (Xj )
and E (Xj Xk ) for j = k . Are Xj a