Download Sample Mean and Law of Large Numbers Consider a finite number

Sample Mean and Law of Large Numbers
Consider a finite number of random variable X, Y , . . ., Z on a sample space S. They are said to be independent
if, for any values xi, yj, . . . , zk,
P (X = xi, Y = yj, . . . , Z = zk) ≡ P (X = xi)P (Y
= yj) . . . P (Z = zk)
In particular, X and Y are independent if
P (X = xi, Y = yj) ≡ P (X = xi)P (Y = yj)
Now let X be a random variable with mean µ. We can consider the numerical outcome of each of
n
independent trials to be a random variable with the same distribution as X. The random variable corresponding
to the ith outcome will be denoted by Xi(i = 1, 2, . . . , n). (We note that the Xiare independent with the same
distribution as X.) The average value of all n outcomes is also a random variable which is denoted by Xnand
called the sample mean. That is:
X1+ X2+ · · · + Xn
n
X n=
The law of large numbers says that as n increases the value of the sample mean X napproaches the mean value µ.
Namely:
Theorem 7.11 (Law of Large Numbers): For any positive number α, no matter how small, the probability that
the sample mean Xnhas a value in the interval [µ − α, µ + α] approaches 1 as n approaches
infinity. That is:
P ([µ − α ≤ X ≤ µ + α]) → 1
as
n → ∞.
x4= 1,
x5= 4
EXAMPLE 7.19 Suppose a die is tossed 5 times with outcomes:
x1= 3,
x2= 4,
x3= 6,
Then the corresponding value x of the sample mean X5follows:
x =3 + 4 + 6 + 1 + 4/5= 3.6
Excercise
SAMPLE SPACES AND EVENTS
7.42. Let A, B, and C be events. Rewrite each of the following events using set notation:
(a) A and B but not C occurs; (c) none of the events occurs;
(b) A or C, but not B occurs; (d) at least two of the events occur.
7.43. A penny, a dime, and a die are tossed.
(a) Describe a suitable sample space S, and find n(S).
(b) Express explicitly the following events:
A = {two heads and an even number}
B = {2 appears}
C = {exactly one head and an odd number}
(c) Express explicitly the events: (i) A and B;
(ii) only B;
(iii) B and C.
FINITE EQUIPROBABLE SPACES
7.44. Determine the probability of each event:
(a) An odd number appears in the toss of a fair die.
(b) One or more heads appear in the toss of four fair coins.
(c) One or both numbers exceed 4 in the toss of two fair dice.
7.45. One card is selected at random from 50 cards numbered 1 to 50. Find the probability that the number on the card is;
(a) greater than 10; (c) greater than 10 and divisible by 5;
(b) divisible by 5;
(d) greater than 10 or divisible by 5.
7.46. Of 10 girls in a class, three have blue eyes. Two of the girls are chosen at random. Find the probability that:
(a) both have blue eyes;
(c) at least one has blue eyes;
(b) neither has blue eyes; (d) exactly one has blue eyes.
7.50. Of 120 students, 60 are studying French, 50 are studying Spanish, and 20 are studying both French and Spanish.
A student is chosen at random. Find the probability that the student is studying: (a) French or Spanish; (b) neither
French nor Spanish; (c) only French; (d) exactly one of the two languages.
FINITE PROBABILITY SPACES
7.51. Decide which of the following functions defines a probability space on S = {a 1, a2, a3}:
1
1
1
(a) P (a1) = 4 , P (a2) = 3 , P (a3) = 2(c)
2
1
2
(b) P (a1) = 3 , P (a2) = − 3 , P (a3) = 3(d)
1
1
1
P (a1) = 6 , P (a2) = 3 , P (a3) = 2
1
2
P (a1) = 0, P (a2) = 3 , P (a3) = 3
7.52. A coin is weighted so that heads is three times as likely to appear as tails. Find P (H ) and P (T ).
7.53. Three students A, B, and C are in a swimming race. A and B have the same probability of winning and each is twice
as likely to win as C. Find the probability that: (a) B wins; (b) C wins; (c) B or C wins.
7.54. Consider the following probability
distribution:
1
2
3
4
5
0.4
0.1
0.1
0.2
Outcome x
Probability P (x) 0.2
Consider the events A = {even number}, B = {2, 3, 4, 5}, C = {1, 2}. Find:
(a) P (A), P (B), P (C); (b) P (A ∩ B), P (A ∩ C), P (B ∩ C).
CONDITIONAL PROBABILITY, INDEPENDENCE
7.56. A fair die is tossed. Consider events A = {2, 4, 6}, B = {1, 2}, C = {1, 2, 3, 4}. Find:
(a) P (A and B) and P (A or C), (c) P (A|C) and P (C|A)
(b) P (A|B) and P (B|A)
(d) P (B|C) and P (C|B)
Decide whether the following are independent: (i) A and B; (ii) A and C; (iii) B and C.
.
7.58. Let A and B be events with P (A) = 0.6, P (B) = 0.3, and P (A ∩ B) = 0.2. Find:
(a) P (A ∪ B); (b) P (A|B); (c) P (B|A).
.
7.60. Let A and B be events with P (A) = 0.3, P (A ∪ B) = 0.5, and P (B) = p. Find p if:
(a) A and B are mutually disjoint;
(b) A and B are independent;
(c) A is a subset of B.
7.61. Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find:
(a) P (A ∩ B) and P (A ∪ B); (b)P (A|B) and P (B|A).
.
7.63. Box A contains six red marbles and two blue marbles, and box B contains two red and four blue. A marble is drawn
at random from each box.
(a) Find the probability p that both marbles are red.
(b) Find the probability p that one is red and one is blue.
.
7.65. Three fair coins are tossed. Consider the events:
A = {all heads or all tails},
B = {at least two heads},
C = {at most two heads}.
Of the pairs (A, B), (A, C), and (B, C), which are independent? Which are dependent?
7.66. Find P (B|A) if: (a) A is a subset of B; (b) A and B are mutually exclusive. (Assume P (A) > 0.)
REPEATED TRIALS, BINOMIAL DISTRIBUTION
7.67. Whenever horses a, b, and c race together, their respective probabilities of winning are 0.3, 0.5, and 0.2. They race
three times.
(a) Find the probability that the same horse wins all three races.
(b) Find the probability that a, b, c each win one race.
7.68. The batting average of a baseball player is 0.300. He comes to bat four times. Find the probability that he will get:
(a) exactly two hits; (b) at least one hit.
.
1
7.70. A certain type of missile hits its target with probability P = 3
(a) If three missiles are fired, find the probability that the target is hit at least once.
(b) Find the number of missiles that should be fired so that there is at least a 90% probability of hitting the target.
RANDOM VARIABLES
7.71. A pair of dice is thrown. Let X denote the minimum of the two numbers which occur. Find the distributions and
expectation of X.
7.72. A fair coin is tossed four times. Let X denote the longest string of heads. Find the distribution and expectation of X.
7.73. A fair coin is tossed until a head or five tails occurs. Find the expected number E of tosses of the coin.
s
7.76. A box contains 10 transistors of which two are defective. A transistor is selected from the box and tested until
a nondefective one is chosen. Find the expected number of transistors
7.78. A player tosses three fair coins. He wins $5 if three heads occur, $3 if two heads occur, and $1 if only one head occurs.
On the other hand, he loses $15 if three tails occur. Find the value of the game to the player.
MEAN, VARIANCE, AND STANDARD DEVIATION
7.79. Find the mean µ, variance σ 2 , and standard deviation σ of each distribution:
(a)
x
f (x)
2
3
8
1/41/21/4
(b)
y
g(y)
−1
0.3
0
0.1
1
0.1
7.80. Find the mean µ, variance σ2, and standard deviation σ
x
f (x)
a
p
2
0.3
3
0.2
of the following two-point distribution where p + q = 1:
b
q
7.81. Let W = XY where X and are the random variables in Problem 7.33. (Recall W (s) =
Find: (a) the distribution h of W ; (b) find E(W ).
Does E(W ) = E(X)E(Y )?
x
−1
1
2
7.82. Let X be a random variable with the
f (x) 0.2 0.5 0.3
distribution:
(a) Find the mean, variance, and standard deviation of X.
(b) Find the distribution, mean, variance, and standard deviation of Y
(i) Y= X4; (ii) Y = 3X.
where:
(XY )(s) =
X(s)Y (s).)
BINOMIAL DISTRIBUTION
7.83. The probability that a women hits a target is p = 1/3. She fires 50 times. Find the expected number µ of times she
will hit the target and the standard deviation σ .
7.84. Team A has probability p = 0.8 of winning each time it plays. Let X denote the number of times A will win in
n = 100 games. Find the mean µ, variance σ2, and standard deviation σ
of X.
.
CHEBYSHEV’S INEQUALITY
7.87. Let X be a random variable with mean µ and standard deviation σ .
Use Chebyshev’s Inequality to estimate P (µ − 3σ
≤ X ≤ µ + 3σ ).
7.88. Let Z be the normal random variable with mean µ = 0 and standard deviation σ
Use Chebyshev’s Inequality to find a value b for which P (−b ≤ Z ≤ b) = 0.9
= 1.