Statistics 3401
Spring 2014 Dr. S.
QUIZ 1
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Print your name above and also somewhere on the back of this quiz. Write your answers directly on this quiz, not on a separate sheet
of paper. You must show adequate amount of work to get
STAT 3401 SOLUTIONS FOR SECTIONS 2.10 AND 2.11
SECTION 2.10
2.124 Let F be the event that a randomly selected voter favors the election issue. Let R be the event that
the voter is a Republican and let D be the event that the voter is a Democrat. The probl
STAT 3401 SOLUTIONS FOR SECTION 3.11
3.167 Y has mean = 11 and standard deviation = 9 = 3 .
a) P ( 6 < Y < 16) = P ( 6 < Y < 16 ) = P ( 6 11 < Y < 16 11) = P ( 5 < Y < 5) =
P ( Y < 5) 1 k12 , where 5 = k . So k =
5
=
5
3
and 1 k12 = 1 (5 /13)2 = 1 ( 3 )2
STAT 3401 SOLUTIONS FOR SECTION 3.8
3.121 Y is Poisson with = 2 .
e 2 2 4
a) P (Y = 4) =
.09022 .
4!
b) P (Y 4) = 1 P (Y 3) . But P (Y 3) = p(0) + p(1) + p(2) + p(3) =
e 2 20 e 2 21 e 2 22 e 2 23
+
+
+
0!
1!
2!
3!
= e 2 (1 + 2 + 2 + 8 ) = e 2 ( 19 ) . So
STAT 3401 SOLUTIONS FOR SECTION 3.9
3.149 Suppose that the random variable Y has the mgf m(t ) = (.6et + .4)3 = ( pet + q )n . This is the
mgf of the binomial distribution with n = 3 and p = .6 (and q = .4 ). By the uniqueness property, Y
must have this d
STAT 3401 SOLUTIONS FOR SECTION 3.7
3.103 Let X be the number of non-defective machines out of the five selected machines. Then X is
hypergeometric with N = 10, r = 10 4 = 6 (non-defectives), n = 5 . So
6 4
5 0
6
1
P ( X = 5) = =
=
.02381 .
252 42
10
STAT 3401 SOLUTIONS FOR SECTION 3.6
3.90 Let Y be the number of employees that must be tested in order to find three that test positive
for asbestos in their lungs. Then Y is negative binomial with r = 3 , p = .40 , and q = 1 p = .60 . So
9
P (Y = 10) =
STAT 3401 SOLUTIONS FOR SECTION 3.5
3.67 Let Y be the number of applicants that are interviewed until the first one is found who has
advanced training in programming. Then Y is geometric with p = .30 and q = 1 p = .7 . So
P (Y = 5) = q 51p = (.7)4 (.3) =
STAT 3401 SOLUTIONS FOR SECTION 3.4
3.39 Let Y be the number of components out of the four components that operate longer than 1000
hours. Then Y is binomial with n = 4 and p = 1 .2 = .8 .
4
a) P (Y = 2) = (.8)2 (.2)2 = 6(.8)2 (.2)2 = .1536 .
2
b) The s
STAT 3401 SOLUTIONS FOR SECTION 3.2
3.1 Let A be the event that impurity A is found in the well and B, the event that impurity B is found.
The problem gives P ( A) = .40, P (B) = .50, P ( A B ) = .20 . By DeMorgans rule and the complement
rule, P ( A B) =
STAT 3401 SOLUTIONS FOR SECTION 2.9
2.110 Let D be the event that a randomly selected item is defective. Let I and II be the events that
correspond to which line the item came from. The problem gives: P ( I) = .40, P ( II) = .60, P (D | I) = .08, and
P (D
STAT 3401 SOLUTIONS FOR SECTION 2.8
2.85 Since, A and B are independent, P ( A B) = P ( A) P (B) and so it follows that
P ( A B) = P ( A) P ( A B) = P ( A) P ( A) P (B) = P ( A) (1 P (B) = P ( A) P (B) . Therefore, A and B are
independent. Another way: By
STAT 3401 SOLUTIONS FOR SECTION 2.6
2.35 Consider the 2-stage job: In stage 1, you select a flight from New York to California; in stage 2, you
select a flight from California to Hawaii. The job can be done in 6 7 = 42 ways.
2.41 Consider the 7-stage job:
STAT 3401 SOLUTIONS FOR SECTION 2.7
2.71
P ( AB) .1 1
=
= .
P ( B)
.3 3
P (BA) .1 1
=
= .
P (B | A) =
.5 5
P ( A)
First note that P ( A B) = P ( A) + P (B) P ( AB) = .5 + .3 .1 = .7 . Also note that A ( A B) = A since
P ( A ( A B)
P ( A)
.5 5
A ( A B) . T
STAT 3401 SOLUTIONS FOR SECTION 2.5
2.26 Label the four cans as: 1, 2, 3, 4. Let the labels 1 and 2 represent the cans with water.
= cfw_ cfw_1,2,cfw_1,3,cfw_1, 4,cfw_2,3,cfw_2, 4,cfw_3, 4 . Each outcome represents the two cans that the expert says
a)
co
STAT 3401 SOLUTIONS FOR SECTION 2.4
2.12 Let L, R, and S be the events that the vehicle turns left, turns right, and goes straight, respectively.
a)
= cfw_L, R, S .
b) P ( It turns ) = P ( cfw_L, R ) =
2
3
.
2.15
a) P (E1 ) + P (E3 ) + P (E4 ) = .01 + .09
STAT 3401 SOLUTIONS FOR SUPPLEMENTARY EXERCISES
2.146
The sample space is the set of all combinations of five cards chosen from the 52 cards and the number of
P
52
such combinations is = 52 5 = 2,598,960 . There are four suits: spades, clubs, diamonds,
STAT 3401 SOLUTIONS FOR SECTION 2.3
2.1
A = cfw_FF . B = cfw_MM . C = cfw_FM , MF, MM . A B = . A B = cfw_FF, MM . A C = .
A C = S. B C = B. B C = C. C B = cfw_FM , MF .
2.2
a) AB = A B .
b) A B .
c) AB = A B .
d) ( AB ) (BA) .
2.7 First note that the sam
GEOMETRIC AND NEGATIVE BINOMIAL STATISTICS 3401
GEOMETRIC DISTRIBUTION:
Consider a sequence of independent Bernoulli trials with constant probability of success p, 0 < p < 1 . Let X
denote the number of trials until the first success. Then X is said to ha
HYPERGEOMETRIC DISTRIBUTION STATISTICS 3401
Suppose that a population consists of N objects each of which is categorized as success (S) or failure (F).
Assume that r objects are successes and the other N r objects are failures. Consider the random
experim
BERNOULLI AND BINOMIAL STATISTICS 3401
The binomial, geometric, and negative binomial distributions are defined in terms of a sequence of
independent Bernoulli trials.
Definition: A Bernoulli trial is a trial of a random experiment that has only two possi
CONVERGENCE IN DISTRIBUTION AND THE CENTRAL LIMIT THEOREM STAT 3401/6204
Definition: Suppose that X1 , X 2 , X 3 , is an infinite sequence of random variables and FXn ( x ) is the cdf of
X n for each n = 1, 2,3, . Further suppose that X is a random variab
MGFS FOR CONTINUOUS DISTRIBUTIONS STATISTICS 3401
Recall that the moment-generating function (mgf) of a random variable X is the function m(t ) = E (e tX ) . For a
continuous X having density function f(x), the LOTUS implies that m (t ) = etx f ( x ) dx .
RANDOM VARIABLES AND THEIR DISTRIBUTIONS STATISTICS 3401
A random variable is a variable that represents a numerical characteristic of the outcomes of a random
experiment. In mathematical terms, a random variable is a rule that assigns a real number to ea
THE POISSON DISTRIBUTION STATISTICS 3401
The Poisson distribution is named after Simeon Poisson who, in 1837, used the distribution as an
approximation to the binomial distribution. The Poisson random variable has the countably infinite range
cfw_ 0, 1, 2
THE NORMAL DISTRIBUTION STATISTICS 3401
The normal distribution is the most important distribution in all of statistics and probability. The normal density function
describes the classic symmetric, bell-shaped curve. One of the first applications of the n
MOMENT-GENERATING FUNCTION STATISTICS 3401/4412
Definition: The moments (about the origin) of a random variable X are the following expected values:
E ( X ), E ( X 2 ), E ( X 3 ), E ( X 4 ), . . For each positive integer k, let = E ( X k ) , the kth momen
INDEPENDENCE OF RANDOM VARIABLES
Recall that events A and B are independent if and only if P ( A B) = P ( A)P (B) . This is the basis for the
definition of independence of random variables.
Definition: Random variables X and Y, defined on the same sample