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Medea of Convergence 81
Theorem 21 CONTINUOUS MAPPING THEOREM Ian L n a constant and
g() to a continuous function at a, then
9(Tn) $9046)
Proof. Let c > 0. By the continuity of g at a, El 7) :> 0 suc
64 Point and Interval Estimation
to dene and nd a best estimator among many possible estimators.
6.1.] Methods of Finding Estimators
Any statistic (known function of observable random variables that i
70 Point and Interval Estimation
means that the actual cost will lie somewhere between 78.5% and 87.5% with high
probability. Let us consider a particular example. Suppose that a random sample
(1.2, 3
82 Modes of Convergence
for all j = 1., .j k. The if part is no surprise and follows from the continuous mapping
theorem. rThe only if part follovvs as if HT.l T < s then Tnj le < n for each j
and so
Testing Proportions 75
Since Z is less than 2.58, reject Hg.
To nd the appropriate test for the mean we have to consider the following
cases:
1. Normal population and known population variance (or sta
76 Hypothesis testing
p: % 0.63 52 0. 6(1 0. 6) 0.24. The Hg : 71' = so and the alternative
is H1 : as > erg. The critical value is 1.64. Now Z = 3%? M = 1.22.
E _ ease/m
Consequently, the null is not
Chapter 9
ASYMPTOTIC THEORY 2
We can now establish the convergence in distribution and the central limit theorem,
which is of great importance.
Denition A sequence of random variables T1, T2, converge
Chapter 8
MODES OF CONVERGENCE
We have a statistic T which is a measurable function of the data
Tn = T(X1= "'an):
and we would like to know what happens to T. as n as oo. It turns out that the limit
i
Sampiing from the Normai Distribution 61
Theorem 15 If the random variables Xi, i = 1,2,.1: are normally and indepen
dently distributed with means in and variances of then
it 2
U : 21(Xi02lbi)
has a c
Interval Estimation 69
the logarithm of the likelihood function is
n n 1 n
log L(p. 02) : log 271' 5 log (.72 27.2 ZW. M2
i=1
To nd the maximum with respect to ,u, and 02 we compute
310gL 1 n
a ;( u)
Chapter 7
HYPOTHESIS TESTING
A statistical hypothesis is an assertion or conjecture, denoted by H, about a
distribution of one or more random variables. If the statistical hypothesis completely
specie
66 Point and Interval Estimation
And in general we should estimate 3) by 0.25 when :1: = 0 or 1 and by 0.75
when a: = 2 or 3. The estimator may be dened as
0.25 for s: = 0.1
0.75 for 3: = 2. 3
zhf=
Th
86 Asymptotic Theory 2
if and only if the Lindebeig condition
2?:1 LtpiEBn (75 _ H402 dFir (t)
f v 0?
'11
n>oo, each e:>0
is satised.
The key condition for the above CLT is the Lindeberg condition. W
Sampiing from the Normal Distribution 59
2
=Z(Xifn)2+(7nH)
Using the above identity we obtain:
an 2 E [ :09 m2] : E [2 (Xi tr n (Xi if] 2
= LEEUQ m2 nE (in of] = i L; (.72 near (3.75%)] =
= i [no2 n
60 Sampling Theory
(1) From a Theorem above we have that
90:05) = [90): (if/RN?
Now (105(t/n) = exp (int %02t2). Hence (pf (t) = [exp (in% %oz (32)] =
exp (int % (3:) t2), which is the cf of a normal
Means and Variances 57
In particular, if r = l, we get the sample mean, which is usually denoted by 7 cry;
that is:
Also the arm sample central moment (about 7%,), denoted by M, is dened
1 _ .
AzggxX.
74 Hypothesis testing
to be the probability that a Type I error is made. and similarly the size of a Type
II error is dened to be the probability that a Type II error is made.
Signicance level or size
Chapter 6
POINT AND INTERVAL ESTIMATION
The problem of estimation is dened as follows. Assume that some characteristic of
the elements in a population can be represented by a random variable X whose d
62 Sampling Theory
5.3.3 The Stridentt Distribntion
If X is a random variable with density
_r[(i+1)/2] 1 1
WWW f WWW
where F(.) is the gamma function, then X is dened to have a t distribution with
k d
80 Modes of Convergence
I
Note that if T. :> 0,
E T. 2 E T.
s s
so that if HT) > 0, this is sufcient for T.1 i 0.
But the converse of the theorem is not necessarily true. To see this consider
th
Solving differential equations by substitution
Some differential equations can be made simpler by substituting a particular form of
solution. Often you will be given a substitution to try.
In the spec
Cisco Packet Tracer Skills Assessments
Formative and summative
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provide an enriching and
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MANAGEMENT:
A GLOBAL AND ENTREPRENEURIAL PERSPECTIVE
by Weihrich, Cannice, and Koontz
Chapter
1
Management: Science, Theory, and Practice
2008
Chapter 1. Management: Science,
1
After studying this ch
ECE001 ELECTRONIC DEVICES AND CIRCUITS
INTRODUCTION TO SEMICONDUCTOR
The Three Kinds of Formulas
A formula is a rule that relates quantities. The rule may be an equation, an inequality, or other math
Kinematics of a Particle
CHAPTER OBJECTIVES
0 To introduce the concepts of position, displacement, velocity, and
acceleration.
° To study particle motion along a straight line and represent this
mot
Introduction
Contain slides by LeonGarcia
and Widjaja
Communication Services &
Applications
A communication service enables the exchange of
information between users at different locations.
Communica
56 Sampling Theory
to nd out something about a certain target population. It is generally impossible
or impractical to examine the entire population, but one may examine a part of it
(a sample from it
58 Sampling Theory
Theorem 10 Let Xth, .,Xy be a random sample from the density f(.). The
expected ualue of the rF1 sample moment is equal to the rm population moment, i.e.
the rm sample moment is an
Asymptetie Theory 2 85
We can now sppromately calculate for example
P(x/(7u) mom)
P(>10) : J 3; J
g P(Z> gM)
= 1_p(wiz)=l_(\/(1OH)
CLT for nonidentically distributed random variables.
Theorem 25 (Ly
84 Asymptotic Theory 2
Theorem 24 Central Limit Theorem ofLindembergLe'vy. l'.etX1.,X2,.X.1 be 33 d.
with E (Xe) = 13, Var (X.) = 02 < oo. Then
(7e) iN(o.e2)
N(0, 1)
$3

lo
The vector version: Let X
Interval Estimation 71
Let Xth, .,X.1 be a random sample from the density f(i;6). Let T1 =
t1(X1. X2, . X.) be a statistic for which P9[T1 <1 T(t9)] = 7. Then T1 is called a one
sided lower condence