Chapter6_theory.lyx

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\begin_body

\begin_layout Part*
Chapter 6 Large Random Samples
\end_layout

\begin_layout Section*
6.1 Introdution
\end_layout

\begin_layout Section*
6.2 The Law of Large Numbers 
\end_layout

\begin_layout Standard
The average of a random sample of i.i.d.
 random variables is called their sample mean.
 The sample mean is useful for summarizing the information in a random sample
 in much the same way that the mean of a probability distribution summarizes
 the information in the distribution.
 In this section, we present some results that illustrate that connection
 between the sample mean and the expected value of the individual random
 variables that comprise the random sample.
\end_layout

\begin_layout Subsection*
The Markov and Chebyshev Inequalities 
\end_layout

\begin_layout Standard

\series bold
Theorem 6.2.1 
\series default
Markov Inequality.
 Suppose that 
\begin_inset Formula $X$
\end_inset

 is a random variable such that 
\begin_inset Formula $Pr(X\geq0)=1$
\end_inset

.
 Then for every real number 
\begin_inset Formula $t>0$
\end_inset

, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(X\geq t)\le\frac{E(X)}{t}
\]

\end_inset

(6.2.1)
\end_layout

\begin_layout Paragraph*

\series medium
The Chebyshev inequality is related to the idea that the variance of a random
 variable is a measure of how spread out its distribution is.
 The inequality says that the probability that 
\begin_inset Formula $X$
\end_inset

 is far away from its mean is bounded by a quantity that 
\series default
increase as 
\begin_inset Formula $Var(X)$
\end_inset

 increases.
\end_layout

\begin_layout Paragraph

\series bold
Theorem 6.2.2 Chebyshev Inequality
\series default
.
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable for which 
\begin_inset Formula $Var(X)$
\end_inset

 exists.
 Then for every number 
\begin_inset Formula $t>0$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(|X-E(X)|\geq t)\leq\frac{Var(X)}{t^{2}}
\]

\end_inset

(6.2.3)
\end_layout

\begin_layout Standard
It can be seen from this proof that the Chebyshev inequality is simply a
 special case of the Markov inequality.
 Therefore, the comments that were given following the proof of the Markov
 inequality canbe applied as well to the Chebyshev inequality.
 Because of their generality, these inequalities are very useful.
 For example, if 
\begin_inset Formula $Var(X)=\sigma^{2}$
\end_inset

 and we let 
\begin_inset Formula $t=3\sigma$
\end_inset

, then the Chebyshev inequality yields the result that
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
P(|X-E(X)|\geq3\sigma)\leq\frac{1}{9}
\]

\end_inset


\end_layout

\begin_layout Standard
In words, the probability that any given random variable will differ from
 its mean by more than 3 standard deviation cannot exceed 1/9.
 This probability will actually be much smaller than 1/9 for many of the
 random variables and distributions that will be discussed in this book.
 The Chebyshev inequality is useful because of the fact that this probability
 must be 1/9 or less for every distribution.
 It can also be shown that the upper bound in (6.2.3) is sharp in the sense
 that it cannot be made any smaller and still hold for all distributions.
\end_layout

\begin_layout Subsection*
Properties of the Sample Mean 
\end_layout

\begin_layout Standard

\series bold
Theorem 6.2.3 
\series default
Mean and Variance of the Sample Mean.
 Let 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 be a random sample from a distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

.
 Let 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 be the sample mean.
 Then 
\begin_inset Formula $E(\bar{X_{n}})=\mu$
\end_inset

 and 
\begin_inset Formula $Var(\bar{X_{n}})=\sigma^{2}/n$
\end_inset


\end_layout

\begin_layout Standard
In words, the mean of 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 is equal to the mean of the distribution from which the random sample was
 drawn, but the variance of 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 is only 
\begin_inset Formula $1/n$
\end_inset

 times the variance of that distribution.
 
\series bold
It follows that the probability distribution of 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

will be more concentrated around the man value 
\begin_inset Formula $\mu$
\end_inset

 than was the original distribution.
 
\series default
In other words, the sample mean 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 is more likely to be close to 
\begin_inset Formula $\mu$
\end_inset

 than is the value of just a single observation 
\begin_inset Formula $X_{i}$
\end_inset

 from the given distribution.
\end_layout

\begin_layout Standard
These statements can be made more precise by applying Chebyshev inequality
 to 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

.
 Since 
\begin_inset Formula $E(\bar{X_{n}})=\mu$
\end_inset

 and 
\begin_inset Formula $Var(\bar{X_{n}})=\sigma^{2}/n$
\end_inset

, it follows from the relation (6.2.3) that for every number 
\begin_inset Formula $t>0$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(|\bar{X_{n}}-\mu|\geq t)\le\frac{\sigma^{2}}{nt^{2}}
\]

\end_inset


\end_layout

\begin_layout Standard
It should be emphasized that the use of the Chebyshev inequality in Example
 6.2.1 gaurantees that samples for which 
\begin_inset Formula $n=400$
\end_inset

 will be large enough to meet the specified probability requirements, regardless
 of the particular type of distribution from which the sample is to be taken.
 It further information about this distribution is available, then it can
 often be shown that a smaller value for 
\begin_inset Formula $n$
\end_inset

 will be sufficient.
 
\end_layout

\begin_layout Subsection*
The law of Large Numbers 
\end_layout

\begin_layout Standard
The discussion in Example 5.2.3 indicates that the Chebyshev inequality may
 not be a practical tool for determining the appropriate sample size in
 a particular problem, because it may specify a much greater sample size
 than is actually needed for the particular distribution from which the
 sample is being taken.
 However, the Chebyshev inequality is a valuable theoretical tool, and it
 will be used here to prove an important result known as the 
\series bold
law of large numbers.
\end_layout

\begin_layout Standard
Suppose that 
\begin_inset Formula $Z_{1},Z_{2},...$
\end_inset

 is a sequence of random variables.
 Roughly speaking, it is said that this sequence converges to a given number
 
\begin_inset Formula $b$
\end_inset

 if the probability distribution of 
\begin_inset Formula $Z_{n}$
\end_inset

 becomes more and more concentrated around 
\begin_inset Formula $b$
\end_inset

 as 
\begin_inset Formula $n\rightarrow\infty$
\end_inset

.
 To be more precise, we give the following definition.
\end_layout

\begin_layout Paragraph
Definition 6.2.1 
\series medium
Convergence in Probability.
 A sequence 
\begin_inset Formula $Z_{1},Z_{2},...$
\end_inset

 of random variables converges to 
\begin_inset Formula $b$
\end_inset

 in probability if for every number 
\begin_inset Formula $\epsilon>0$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
lim_{n\rightarrow\infty}Pr(|Z_{n}-b|<\epsilon)=1
\]

\end_inset


\end_layout

\begin_layout Standard
This property is denote by 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Z_{n}\underrightarrow{p}b
\]

\end_inset


\end_layout

\begin_layout Standard
and is sometimes stated simply as 
\begin_inset Formula $Z_{n}$
\end_inset

 converges to 
\begin_inset Formula $b$
\end_inset

 in pribability.
\end_layout

\begin_layout Standard

\series bold
In other words, 
\begin_inset Formula $Z_{n}$
\end_inset

 converges to 
\begin_inset Formula $b$
\end_inset

 in probability if the probability that 
\begin_inset Formula $Z_{n}$
\end_inset

 lies in each given interval around 
\begin_inset Formula $b$
\end_inset

, no matter how small this interval maybe, approaches 
\begin_inset Formula $1$
\end_inset

 as 
\begin_inset Formula $n\rightarrow\infty$
\end_inset


\series default
.
\end_layout

\begin_layout Paragraph
Theorem 6.2.4 Law of Large Numbers
\series medium
.
 Suppose that 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 form a random sample from a distribution for which the mean is 
\begin_inset Formula $\mu$
\end_inset

 and for which the variance is finite.
 Let 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 denote the sample mean.
 Then
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\bar{X_{n}}\underrightarrow{p}\mu
\]

\end_inset


\end_layout

\begin_layout Standard
Since 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 converges to 
\begin_inset Formula $\mu$
\end_inset

 in probability, it follows that there is high probability that 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 will be close to 
\begin_inset Formula $\mu$
\end_inset

 if the sample size 
\begin_inset Formula $n$
\end_inset

 is large.
 Hence, if a large random sample is taken from a distribution for which
 the mean is unknown, then the arithmetic average of the values in the sample
 will usually be a close estimate of the unknown mean.
 This topic will be discussed again in Sec.
 6.3, where we introduce the central limit theorem.
 It will then be possible to present a more precise probability distribution
 for the difference between 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

and 
\begin_inset Formula $\mu$
\end_inset

.
\end_layout

\begin_layout Standard
The following result can be useful if we observe random variables with mean
 
\begin_inset Formula $\mu$
\end_inset

 but are interested in 
\begin_inset Formula $\mu^{2}$
\end_inset

 or 
\begin_inset Formula $log(\mu)$
\end_inset

 or some other continuous function of 
\begin_inset Formula $\mu$
\end_inset

.
\end_layout

\begin_layout Paragraph
Theorem 6.2.5 
\series medium
Continuous Functions of Random Variables.
 If 
\begin_inset Formula $Z_{n}\rightarrow_{p}b$
\end_inset

, and if 
\begin_inset Formula $g(z)$
\end_inset

 is a function that is continuous at 
\begin_inset Formula $z=b$
\end_inset

, then 
\begin_inset Formula $g(Z_{n})\rightarrow_{p}g(b)$
\end_inset

.
\end_layout

\begin_layout Standard
Theorem 6.2.5 extends to any finite number 
\begin_inset Formula $k$
\end_inset

 of sequences that converge in probabiliy and a continuous function of 
\begin_inset Formula $k$
\end_inset

 variables.
\end_layout

\begin_layout Standard
The law of large numbers helps to explain why a histogram (Definition 3.7.9)
 can be used as an approximation to an p.d.f.
\end_layout

\begin_layout Paragraph
Theorem 6.2.6 
\series medium
Histograms.
 Let 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 be a sequence of i.i.d.
 random variables.
 Let 
\begin_inset Formula $c_{1}<c_{2}$
\end_inset

 be two constants.
 Define 
\begin_inset Formula $Y_{i}=1$
\end_inset

 if 
\begin_inset Formula $c_{1}\leq X_{i}\leq c_{2}$
\end_inset

 and 
\begin_inset Formula $Y_{i}=0$
\end_inset

 if not.
 Then 
\begin_inset Formula $\bar{Y_{n}}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}$
\end_inset

 is the proportion of 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 that lie in the interval 
\begin_inset Formula $[c_{1},c_{2})$
\end_inset

, and 
\begin_inset Formula $\bar{Y_{n}}\rightarrow_{p}Pr(c_{1}\leq X_{1}\leq c_{2})$
\end_inset


\end_layout

\begin_layout Standard
In words, Theorem 6.2.6 says the following: If we draw a histogram with the
 area of the bar over each subinterval being the proportion of a random
 sample that lies in the corresponding subinterval, then the area of each
 bar converges in probability to the probability that a random variable
 from the sequence lies in the subinterval, then the area of each bar converges
 in probability to the probability that a random variable from the sequence
 lies in the subinterval.
 If the sample is large, we would then expect the area of each bar to be
 close to the probability.
 The same idea applies to a conditionally i.i.d.
 (given 
\begin_inset Formula $Z=z$
\end_inset

) sample, with 
\begin_inset Formula $Pr(c_{1}\leq X_{1}\leq c_{2})$
\end_inset

 replaced by 
\begin_inset Formula $Pr(c_{1}\leq X_{1}\leq c_{2}|Z=z)$
\end_inset

.
\end_layout

\begin_layout Subsection*
Weak Laws and Strong Laws
\end_layout

\begin_layout Standard
Other concepts of the convergence of a sequence of random variables:
\end_layout

\begin_layout Standard
For example, it is said that a sequence 
\begin_inset Formula $Z_{1},Z_{2},...$
\end_inset

 
\emph on
converges to 
\begin_inset Formula $a$
\end_inset

 constant 
\begin_inset Formula $b$
\end_inset

 with probability 
\begin_inset Formula $1$
\end_inset

 if 
\begin_inset Formula $Pr(\lim_{n\rightarrow\infty}(Z_{n}=b))=1(1)$
\end_inset

 (
\emph default
It is different with 
\begin_inset Formula $\lim_{n\rightarrow\infty}(Pr(Z_{n}=b))=1(2)$
\end_inset


\end_layout

\begin_layout Standard
It can be shown that if a sequence 
\begin_inset Formula $Z_{1},Z_{2},...$
\end_inset

 converges to 
\begin_inset Formula $b$
\end_inset

 with probability 
\begin_inset Formula $1$
\end_inset

 (1), then the sequence will also converge to 
\begin_inset Formula $b$
\end_inset

 in probability (2).
 For this reason, convergence with probability 
\begin_inset Formula $1$
\end_inset

 is often called 
\series bold
\emph on
strong convergence
\series default
, 
\emph default
whereas convergence in probability is called 
\series bold
weak convergence
\series default
 .
 In order to emphasize the distinction between these two concepts of convergence
, the result that here has been called simply the law of large numbers is
 often called the 
\series bold
weak law of large numbers.
\end_layout

\begin_layout Paragraph
The Strong law of large numbers: 
\series medium
If 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 is the sample mean of a random sample of size 
\begin_inset Formula $n$
\end_inset

 from a distribution with mean 
\begin_inset Formula $\mu$
\end_inset

, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(\lim_{n\rightarrow\infty}\bar{X_{n}}=\mu)=1
\]

\end_inset


\end_layout

\begin_layout Standard
There are examples of sequences of random variables that converge in probability
 but that do not converge with probability 1.
\end_layout

\begin_layout Standard
The choice of 
\begin_inset Formula $s=1/2$
\end_inset

 in Example 6.2.6 was arbitrary.
 Theorem 6.2.7 says that we can replace this arbitrary choice with the choice
 that leads to the smallest possible bound.
 
\end_layout

\begin_layout Paragraph
Theorem 6.2.7 
\series medium
Chernoff Bounds.
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable with moment generating function 
\begin_inset Formula $\psi$
\end_inset

.
 Then for every real 
\begin_inset Formula $t$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(X\geq t)\leq\min_{s>0}exp(-st)\psi(s)
\]

\end_inset


\end_layout

\begin_layout Standard
Theorem 6.2.7 is most useful when 
\begin_inset Formula $X$
\end_inset

 is the sum of 
\begin_inset Formula $n$
\end_inset

 i.i.d.
 random variabels each with finite m.g.f and when 
\begin_inset Formula $t=nu$
\end_inset

 for a large value of 
\begin_inset Formula $n$
\end_inset

 and some fixed 
\begin_inset Formula $u$
\end_inset

.
 This was the case in Example 6.2.6.
\end_layout

\begin_layout Paragraph
Summary 
\end_layout

\begin_layout Standard
The law of large numbers says that the sample mean of a random sample converges
 in probability to the mean 
\begin_inset Formula $\mu$
\end_inset

 of the individual random variables, if the variance exists.
 This means that the sample mean will be close to 
\begin_inset Formula $\mu$
\end_inset

 if the size of the random sample is sufficiently large.
 The Chebyshev inequality provides a (crude) bound on how hight the probability
 is that the sample mean will be close to 
\begin_inset Formula $\mu$
\end_inset

.
 
\series bold
Chernoff bounds 
\series default
can be sharper, but are harder to compute.
\end_layout

\begin_layout Section*
6.3 The Central Limit Theorem 
\end_layout

\begin_layout Standard
The sample mean of a large random sample of random variables with mean 
\begin_inset Formula $\mu$
\end_inset

 and finite variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

 has approximately the normal distribution with eman 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}/n$
\end_inset

.
 This result helps to justify the use of the normal distribution as model
 for many random variables that can be thought of as being made up of many
 independent random variables that are not identically distributed.
 We also introduce the delta method, which allows us to compute approximate
 distributions for functions of random varibles.
 
\end_layout

\begin_layout Paragraph
Statement of the Theorem 
\end_layout

\begin_layout Standard
In examples 5.4.1 and 5.4.2, we illustrated how the Poisson distribution provides
 a good approximation to binomial distribution with a large 
\begin_inset Formula $n$
\end_inset

 and small 
\begin_inset Formula $p$
\end_inset

.
 Example 6.3.1 shows how a normal distribution can be a good approximation
 to a binomial distribution with a large 
\begin_inset Formula $n$
\end_inset

 and not so small 
\begin_inset Formula $p$
\end_inset

.
 The central limit theorem (Theorem 6.3.1) is a formal statement of how normal
 distributions can approximate distributions of general sums or averages
 of i.i.d.
 random variables.
\end_layout

\begin_layout Standard
In Corollary 5.6.2, we saw that if a random sample of size 
\begin_inset Formula $n$
\end_inset

 is taken from the normal distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

, then the sample average 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 has the normal distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}/n$
\end_inset

.
 The simple version of the central limit theorem that we give in this section
 says that whenever a random sample of size 
\begin_inset Formula $n$
\end_inset

 is taken from any distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

, the sample average 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 will have a distribution that is approximately normal with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}/n$
\end_inset

.
\end_layout

\begin_layout Standard
Authors shall also state another central limit theorem pertaining to the
 sum of independent random variables that are not necessarily identically
 distributed and shall present some examples illustrating both theorems.
\end_layout

\begin_layout Paragraph
Theorem 6.3.1 
\series medium
Central Limit Theorem (Lindeberg and Levy).
 If the random variables 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 form a random sample of size 
\begin_inset Formula $n$
\end_inset

 from a given distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}\text{ }(0<\sigma^{2}<\infty)$
\end_inset

, then for each fixed number 
\begin_inset Formula $x$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lim_{n\rightarrow\infty}Pr[\frac{\bar{X_{n}}-\mu}{\sigma/n^{1/2}}\le x]=\Phi(x)\text{ (6.3.1)}
\]

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $\Phi$
\end_inset

 denotes the c.d.f.
 of the standard normal distribution.
\end_layout

\begin_layout Standard
The interpretation of Eq.
 (6.3.1) is as follows: If a large random sample is taken from any distribution
 with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

, regardless of whether this distribution is discrete or continuous, then
 the distribution of random variable 
\begin_inset Formula $n^{1/2}(\bar{X_{n}}-\mu)/\sigma$
\end_inset

 will be approximately the standard normal distribution.
 Therefore, the distribution of 
\begin_inset Formula $\bar{X_{n}}$
\end_inset

 will be approximately the normal distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}/n$
\end_inset

, or, equivalently, the distribution of the sum 
\begin_inset Formula $\sum_{i=1}^{n}X_{i}$
\end_inset

 will be approximately the normal distribution with mean 
\begin_inset Formula $n\mu$
\end_inset

 and variance 
\begin_inset Formula $n\sigma^{2}$
\end_inset

.
 
\end_layout

\begin_layout Paragraph
Definition 6.3.1 
\series medium
Convergence in Distribution/Asymptotic Distribution.
 Let 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 be a sequence of random variables, and for 
\begin_inset Formula $n=1,2,...$
\end_inset

, let 
\begin_inset Formula $F_{n}$
\end_inset

 denote the c.d.f.
 of 
\begin_inset Formula $X_{n}$
\end_inset

.
 Also, let 
\begin_inset Formula $F^{*}$
\end_inset

 be a c.d.f.
 Then it is said that the sequence 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 converges in distribution to 
\begin_inset Formula $F^{*}$
\end_inset

 if 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lim_{n\rightarrow\infty}F_{n}(x)=F^{*}(x)\text{ (6.3.3)}
\]

\end_inset


\end_layout

\begin_layout Standard
For all 
\begin_inset Formula $x$
\end_inset

 at which 
\begin_inset Formula $F^{*}(x)$
\end_inset

 is continuous.
 Sometimes, it is simply said that 
\begin_inset Formula $X_{n}$
\end_inset

 converges in distribution to 
\begin_inset Formula $F^{*}$
\end_inset

, and 
\begin_inset Formula $F^{*}$
\end_inset

 is called the asymptotic distribution of 
\begin_inset Formula $X_{n}$
\end_inset

.
 If 
\begin_inset Formula $F^{*}$
\end_inset

 has a name, then we say that 
\begin_inset Formula $X_{n}$
\end_inset

 converges in distribution to that name.
\end_layout

\begin_layout Standard
Thus, according to Theorem 6.3.1, as indicated in Eq.
 (6.3.1), the random variable 
\begin_inset Formula $n^{1/2}(\bar{X_{n}}-\mu)/\sigma$
\end_inset

 converges in distribution to the standard normal distribution, or, equivalently
, the asymptotic distribution of 
\begin_inset Formula $n^{1/2}(\bar{X_{n}}-\mu)/\sigma$
\end_inset

 is the standard normal distribution.
\end_layout

\begin_layout Paragraph
Effect of the Central Limit Theorem 
\series medium
The central limit theorem provides a plausible explanation for the fact
 that distributions of many random variables studied in phydical experiments
 are approximately normal.
 For example, a person's height is influenced by many random factors.
 IF the height of each person is determined by adding the values of these
 individual factors, then the distribution of the heights of a large number
 of persons will be approximately normal.
 In general, the central limit theorem indicates that the distribution of
 the sum of many random variables can be approximately normal, even though
 the distribution of each random variable in the sum differs from the normal.
\end_layout

\begin_layout Paragraph
Other Examples of Convergence in Distribution: 
\series medium
In chapter 5, we saw three examples of limit theorems involving discrete
 distributions: Theorem 5.3.4, 5.4.5 and 5.4.6 all showed that a sequence of p.f;s
 converged to some other p.f.
 (Binomial ~ Poisson, ..)
\end_layout

\begin_layout Section*
The Delta Method 
\end_layout

\begin_layout Standard
Suppose that 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 form a random sample from a distribution that has finite mean 
\begin_inset Formula $\mu$
\end_inset

 and finite variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

.
 The central limit theorem says that 
\begin_inset Formula $n^{1/2}(\bar{X_{n}}-\mu)/\sigma$
\end_inset

 has approximately the standard normal distribution.
 Now suppose that we are interested in the distribution of some function
 
\begin_inset Formula $\alpha$
\end_inset

 is a differentiable function whose derivative is nonzero at 
\begin_inset Formula $\mu$
\end_inset

.
 We shall approximate the distribution of 
\begin_inset Formula $\alpha(\bar{X_{n})}$
\end_inset

 by a method known in statistics as the 
\emph on
delta method.
\end_layout

\begin_layout Paragraph
Theorem 6.3.2 
\series medium
Delta Method.
 Let 
\begin_inset Formula $Y_{1},Y_{2},...$
\end_inset

 be a sequence of random variables, and let 
\begin_inset Formula $F^{*}$
\end_inset

 be a continuous c.d.f.
 Let 
\begin_inset Formula $\theta$
\end_inset

 be a real number, and let 
\begin_inset Formula $a_{1},a_{2},..$
\end_inset

 be sequence of positive numbers that increase to 
\begin_inset Formula $\infty$
\end_inset

, Suppose that 
\begin_inset Formula $a_{n}(Y_{n}-\theta)$
\end_inset

 converges in distribution to 
\begin_inset Formula $F^{*}$
\end_inset

.
 Let 
\begin_inset Formula $\alpha$
\end_inset

 be a function with continuous derivative such that 
\begin_inset Formula $\alpha^{'}(\theta)\neq0$
\end_inset

.
 Then 
\begin_inset Formula $a_{n}[\alpha(Y_{n})-\alpha(\theta)]/\alpha^{'}(\theta)$
\end_inset

 converges in distribution to 
\begin_inset Formula $F^{*}$
\end_inset

.
\end_layout

\begin_layout Standard
The most common application of Theorem 6.3.2 occurs when 
\begin_inset Formula $Y_{n}$
\end_inset

 is the average of a random sample from a distribution with finite variance.
 We state that case in the following corollary.
\end_layout

\begin_layout Paragraph
Corollary 6.3.1 
\series medium
Delta Method for Average of a Random Sample.
 Let 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 be a sequence of i.i.d.
 random variables from a distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and finite variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

.
 Let 
\begin_inset Formula $\alpha$
\end_inset

 be a function with continuous derivative such that 
\begin_inset Formula $\alpha^{'}(\mu)\neq0$
\end_inset

.
 Then the asymptotic distribution of 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\frac{n^{1/2}}{\sigma\alpha^{'}(\mu)}[\alpha(\bar{X_{n}})-\alpha(\mu)]
\]

\end_inset

 is the standard normal distribution.
\end_layout

\begin_layout Standard
A common way to report the result in Corollary 6.3.1 is to say that the distributi
on of 
\begin_inset Formula $\alpha(\bar{X_{n}})$
\end_inset

 is approximately the normal distribution with mean 
\begin_inset Formula $\alpha(\mu)$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}[\alpha^{'}(\mu)]^{2}/n$
\end_inset

.
\end_layout

\begin_layout Paragraph
Variance Stabilizing Transformations 
\series medium
If we were to observe a random sample of Possion random variables as in
 Example 6.3.4, we would assume that 
\begin_inset Formula $\theta$
\end_inset

 us unknown.
 In such a case we cannot compute the probability in Eq.
\end_layout

\begin_layout Subsection*
The Central Limit Thoerem (Liapounov) for the Sum of Independent Random
 Variables
\end_layout

\begin_layout Standard
We shall now state a central limit theorem that applies to a sequence of
 random variables 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 that are independent but 
\series bold
not necessarily
\series default
 
\series bold
identically distributed
\series default
.
 This theorem was first proved by A.Liapounov in 1901.
 We shall assume that 
\begin_inset Formula $E(X_{i})=\mu_{i}$
\end_inset

 and 
\begin_inset Formula $Var(X_{i})=\sigma_{i}^{2}$
\end_inset

 for 
\begin_inset Formula $i=1,...,n$
\end_inset

.
 Also, we shall let
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Y_{n}=\frac{\sum_{i=1}^{n}X_{i}-\sum_{i=1}^{n}\mu_{i}}{(\sum_{i=1}^{n}\sigma_{i}^{2})^{1/2}}\text{ (6.3.8)}
\]

\end_inset


\end_layout

\begin_layout Standard
Then 
\begin_inset Formula $E(Y_{n})=0$
\end_inset

 and 
\begin_inset Formula $Var(Y_{n})=1$
\end_inset

.
 The theorem that is stated next gives a sufficient condition for the distributi
on of this random variable 
\begin_inset Formula $Y_{n}$
\end_inset

 to be approximately the standard normal distribution.
\end_layout

\begin_layout Paragraph
Theorem 6.3.3 
\series medium
Suppose that the random varibales 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 are independent and that 
\begin_inset Formula $E(|X_{i}-\mu_{i}|^{3})<\infty$
\end_inset

 for 
\begin_inset Formula $i=1,2,...$
\end_inset

.
 Also, suppose that 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}E(|X_{i}-\mu_{i}|^{3})}{(\sum_{i=1}^{n}\sigma_{i}^{2})^{3/2}}=0\text{ (6.3.9)}
\]

\end_inset


\end_layout

\begin_layout Standard
Finally, let the random variable 
\begin_inset Formula $Y_{n}$
\end_inset

 be as defined in Eq.
 (6.3.8).
 Then, for each fixed number 
\begin_inset Formula $x$
\end_inset

, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lim_{n\rightarrow\infty}Pr(Y_{n}\leq x)=\Phi(x)\text{ (6.3.10)}
\]

\end_inset


\end_layout

\begin_layout Standard
The interpretatino of this theorem is as follows: If Eq.
 (6.3.9) is satisfied, then for every large value of 
\begin_inset Formula $n$
\end_inset

, the distribution of 
\begin_inset Formula $\sum_{i=1}^{n}X_{i}$
\end_inset

 will be approximately the normal distribution with mean 
\begin_inset Formula $\sum_{i=1}^{n}\mu_{i}$
\end_inset

 and variance 
\begin_inset Formula $\sum_{i=1}^{n}\sigma_{i}^{2}.$
\end_inset

 It should be noted that when the random variables 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 are identically distributed and the third moments of the variables exist,
 Eq.
 (6.3.9) will automatically be satisfied and Eq.
 (6.3.10) then reduces to Eq.
 (6.3.1).
\end_layout

\begin_layout Standard
THe distinction between the theorem of Lindeberg and Levy and the theorem
 of Liapounov should be emphasized.
 The theorem of Lindeberg and Levy applies to a sequence of i.i.d random variables.
 In order for this theorem to be applicable, it is sufficient to assume
 only that the variance of each random variable is finite.
 The theorem of Liapounov applies to a sequence of independent random variables
 that are not necessarily identically distributed.
 In order for this theorem to be applicable, it must be assume that the
 third moment of each random varible is finite and satisfied Eq.
 (6.3.9.).
\end_layout

\begin_layout Standard

\series bold
The Central Limit Theorem for Bernoulli Random Variables 
\series default
By applying the theorem of Liapounov, we can establish the following result.
\end_layout

\begin_layout Paragraph
Theorem 6.3.4.
 
\series medium
Suppose that the random variables 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are independent and 
\begin_inset Formula $X_{i}$
\end_inset

 has the Bernoulli distribution with parameter 
\begin_inset Formula $p_{i}(i=1,2,...)$
\end_inset

 .
 Suppose also that the infinite series 
\begin_inset Formula $\sum_{i=1}^{\infty}p_{i}(1-p_{i})$
\end_inset

 is divergent, and let 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Y_{n}=\frac{\sum_{i=1}^{n}X_{i}-\sum_{i=1}^{n}p_{i}}{(\sum_{i=1}^{n}p_{i}(1-p_{i}))^{1/2}}\text{ (6.3.11)}
\]

\end_inset


\end_layout

\begin_layout Standard
Then for every fixed number 
\begin_inset Formula $x$
\end_inset

, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lim_{n\rightarrow\infty}Pr(Y_{n}\leq x)=\Phi(x)\text{ (6.3.12)}
\]

\end_inset


\end_layout

\begin_layout Standard
To prove this, we need to prove its satisfied Eq (6.3.9)
\end_layout

\begin_layout Paragraph

\series medium
Since the normal distribution will be attained more and more closely as
 
\begin_inset Formula $\sum_{i=1}^{n}p_{i}(1-p_{i})\rightarrow\infty$
\end_inset

, the normal distribution provides a good approximiation when the value
 of 
\begin_inset Formula $\sum_{i=1}^{n}p_{i}(1-p_{i})$
\end_inset

 is alrge.
 Furthermore, since the value of each term 
\begin_inset Formula $p_{i}(1-p_{i})$
\end_inset

 is a maximum when 
\begin_inset Formula $p_{i}=1/2$
\end_inset

, the approximation will be best when 
\begin_inset Formula $n$
\end_inset

 is large and the values of 
\begin_inset Formula $p_{1},...,p_{n}$
\end_inset

 are close to 
\begin_inset Formula $1/2$
\end_inset

.
\end_layout

\begin_layout Paragraph
Variance Stabilizing Transformations 
\series medium
If we were to observe a random sample of Poisson random variables as in
 Example 6.3.4, we would assume that 
\begin_inset Formula $\theta$
\end_inset

 is unknown.
 In such a case we cannot compute the probability, because the appronimate
 variance of 
\begin_inset Formula $\bar{X}_{n}$
\end_inset

 depends on 
\begin_inset Formula $\theta$
\end_inset

.
 It is sometimes desirable to transform 
\begin_inset Formula $\bar{X}_{n}$
\end_inset

 by a function 
\begin_inset Formula $\alpha$
\end_inset

 so that the approximate distribution of 
\begin_inset Formula $\alpha(\bar{X}_{n})$
\end_inset

 has variance that is a known value.
 Such a function is called a variance stabilizing transformation.
 We can often find a variance stabilizing transformation by running the
 delta method in reverse.
 In general we note that the approximate distribution of 
\begin_inset Formula $\alpha(\bar{X}_{n})$
\end_inset

 has variacne 
\begin_inset Formula $\alpha^{'}(\mu)^{2}\sigma^{2}/n$
\end_inset

.
 In order to make this variance constant, we need 
\begin_inset Formula $\alpha^{'}(\mu)$
\end_inset

 to be a constant times 
\begin_inset Formula $1/\sigma$
\end_inset

.
 If 
\begin_inset Formula $\sigma^{2}$
\end_inset

 is a function 
\begin_inset Formula $g(\mu)$
\end_inset

, then we achieve this goal by letting 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\alpha(\mu)=\int_{a}^{\mu}\frac{dx}{g(x)^{1/2}}\text{ (6.3.6)}
\]

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $a$
\end_inset

 is an arbitrary constant that makes the integral finite.
\end_layout

\begin_layout Subsection*
Outline of Proof of Central Limit Theorem 
\end_layout

\begin_layout Paragraph
Convergence of the Moment Generating Functions.
 
\series medium
Moment generating functions are important in the study of convergence in
 distribution because 
\end_layout

\begin_layout Paragraph
Theorem 6.3.5 
\series medium
Let 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 be a sequence of random variables.
 For 
\begin_inset Formula $n=1,2,...$
\end_inset

, Let 
\begin_inset Formula $F_{n}$
\end_inset

 denote the c.d.f of 
\begin_inset Formula $X_{n}$
\end_inset

, and let 
\begin_inset Formula $\psi_{n}$
\end_inset

 denote the m.g.f of 
\begin_inset Formula $X_{n}$
\end_inset

.
\end_layout

\begin_layout Standard
Also, Let 
\begin_inset Formula $X^{*}$
\end_inset

 denote another random variable with c.d.f 
\begin_inset Formula $F^{*}$
\end_inset

 and m.g.f.
 
\begin_inset Formula $\psi^{*}$
\end_inset

.
 Suppose that the m.g.f's 
\begin_inset Formula $\psi_{n}$
\end_inset

 and 
\begin_inset Formula $\psi^{*}$
\end_inset

 exists 
\begin_inset Formula $(n=1,2,...)$
\end_inset

.
 If 
\begin_inset Formula $\lim_{n\rightarrow\infty}\psi_{n}(t)=\psi^{*}(t)$
\end_inset

 for all values of 
\begin_inset Formula $t$
\end_inset

 in some interval around the point 
\begin_inset Formula $t=0$
\end_inset

, then the sequence 
\begin_inset Formula $X_{1},X_{2},...$
\end_inset

 converges in distribution to 
\begin_inset Formula $X^{*}$
\end_inset

.
\end_layout

\begin_layout Paragraph
In other words, the sequence of c.d.f's 
\begin_inset Formula $F_{1},F_{2},...$
\end_inset

 must converge to the c.d.f.
 
\begin_inset Formula $F^{*}$
\end_inset

 if the corresponding sequence of m.g.f.'s 
\begin_inset Formula $\psi_{1},\psi_{2},...$
\end_inset

 converges to m.g.f 
\begin_inset Formula $\psi^{*}$
\end_inset

.
\end_layout

\begin_layout Paragraph
Summary
\end_layout

\begin_layout Standard
Two versions of the central limit theorem were given.
 They conclude that the distribution of the average of a large number of
 independent random variables is close to a normal distribution.
 One theorem requires that the random variabels all have the same distribution
 with finite variance.
 The other theorem does not require that the random variabels be identically
 distributed, but instead requires that their third moments exist and satisfy
 condition (6.3.9).
 The delta method lets us find the approximate distribution of a smooth
 function of a sample average.
\end_layout

\begin_layout Section*
6.4 The Correction for Continuity
\end_layout

\begin_layout Standard
Some applications of the central limit theorem allow us to approximate the
 probability that a discrete random variable 
\begin_inset Formula $X$
\end_inset

 lies in an interval 
\begin_inset Formula $[a,b]$
\end_inset

 by the probability that a normal random variable lies in that interval.
 The approximation can be improved slightly by being careful about how we
 approximate 
\begin_inset Formula $Pr(X=a)$
\end_inset

 and 
\begin_inset Formula $Pr(X=b)$
\end_inset

.
\end_layout

\begin_layout Paragraph
Problem : 
\series medium
In Figure 6.4., two c.d.f's are very close at 
\begin_inset Formula $x=n+0.5$
\end_inset

 for each integer 
\begin_inset Formula $n$
\end_inset

.
 But for each integer 
\begin_inset Formula $n$
\end_inset

, 
\begin_inset Formula $Pr(Y\le x)<Pr(X\le x)$
\end_inset

 for 
\begin_inset Formula $x$
\end_inset

 little above 
\begin_inset Formula $n$
\end_inset

 and 
\begin_inset Formula $Pr(Y\leq x)>Pr(X\leq x)$
\end_inset

 for 
\begin_inset Formula $x$
\end_inset

 a little below 
\begin_inset Formula $n$
\end_inset

.
\end_layout

\begin_layout Paragraph
Approximating a Discrete Distribution by a Continous Distribution
\end_layout

\begin_layout Paragraph

\series medium
If he distribution of 
\begin_inset Formula $Y$
\end_inset

 provides a good approximation to the distribution of 
\begin_inset Formula $X$
\end_inset

, then for all integers 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

, we can approximate the discrete probability 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(a\leq X\leq b)=\sum_{x=a}^{b}f(x)\text{ (6.4.1)}
\]

\end_inset


\end_layout

\begin_layout Standard
by the continuous probability 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(a\le Y\leq b)=\int_{a}^{b}g(x)dx\text{ (6.4.2)}
\]

\end_inset


\end_layout

\begin_layout Standard
Indeed, this approximation was used in Examples 6.3.2 and 6.3.9, where 
\begin_inset Formula $g(x)$
\end_inset

 was the appropriate normal p.d.f derived from the central limit theorem.
 
\end_layout

\begin_layout Standard
This simple approximation has the following shortcoming: Although 
\begin_inset Formula $Pr(X\geq a)$
\end_inset

 and 
\begin_inset Formula $Pr(X>a)$
\end_inset

 will typically have different values for the discrete distribution values
 for the discrete distribution of 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $Pr(Y\geq a)=P(Y>a)$
\end_inset

 because 
\begin_inset Formula $Y$
\end_inset

 has a continuous distribution.
 Another way of expressing this shortcoming is as follows: Although 
\begin_inset Formula $Pr(X=x)>0$
\end_inset

 for each integer 
\begin_inset Formula $x$
\end_inset

 that is a possible value of 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $Pr(Y=x)=0$
\end_inset

 for all 
\begin_inset Formula $x$
\end_inset

, because 
\begin_inset Formula $Y$
\end_inset

 has continuous distribution.
\end_layout

\begin_layout Subsection*
Approximating a Bar Chart
\end_layout

\begin_layout Paragraph

\series medium
The p.f.
 
\begin_inset Formula $f(x)$
\end_inset

 of discrete random variable 
\begin_inset Formula $X$
\end_inset

 can be represented by a bar chart, as sketched in Fig.
 6.5.
 For each integer 
\begin_inset Formula $x$
\end_inset

, the probability of 
\begin_inset Formula $\{X=x\}$
\end_inset

 is represented by the area of a rectangle with a base that extends from
 
\begin_inset Formula $x-\frac{1}{2}$
\end_inset

 to 
\begin_inset Formula $x+\frac{1}{2}$
\end_inset

 and with a height 
\begin_inset Formula $f(x)$
\end_inset

.
 Thus, the area of the rectangle for which the center of the base is at
 integer 
\begin_inset Formula $x$
\end_inset

 is simply 
\begin_inset Formula $f(x)$
\end_inset

.
 
\end_layout

\begin_layout Standard
From this point of view, it can be seen that 
\begin_inset Formula $Pr(a\leq X\leq b)$
\end_inset

, as specified in Eq.
 (6.4.1) is the sum of the areas of the rectangles in Fig 6.5 that are centered
 at 
\begin_inset Formula $a,a+1,...,b$
\end_inset

.
 It can also be seen from Fig.
 6.5 that the sum of these areas is approximated by the integral
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(a-1/2<Y<b+1/2)=\int_{a-1/2}^{b+1/2}g(x)dx\text{ (6.4.3)}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename 6.5HistApprox.png

\end_inset


\end_layout

\begin_layout Paragraph

\series medium
The adjustment from the integral in (6.4.2) to the integral in (6.4.3) is called
 the 
\series default
correction for continuity.
\end_layout

\begin_layout Paragraph
Intuition: 
\series medium
Shift place 
\begin_inset Formula $P(Y\leq a+...)$
\end_inset

 or 
\begin_inset Formula $P(Y\leq a-...)$
\end_inset

 to approximate 
\begin_inset Formula $P(X\leq a)$
\end_inset

.
\end_layout

\begin_layout Paragraph

\series medium
If we use the correction for continuity, we find that the probability 
\begin_inset Formula $f(a)$
\end_inset

 of the single integer 
\begin_inset Formula $a$
\end_inset

 can be approximated as follows: 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(X=a)=Pr(a-\frac{1}{2}\leq X\leq a+\frac{1}{2})\approx\int_{a-1/2}^{a+1/2}g(x)dx.\text{ (6.4.4)}
\]

\end_inset


\end_layout

\begin_layout Standard
Similarly, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Pr(X>a)=Pr(X\geq a+1)=Pr(X\geq a+\frac{1}{2})\approx\int_{a+1/2}^{\infty}g(x)dx\text{ (6.4.5)}
\]

\end_inset


\end_layout

\begin_layout Paragraph
Summary
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $X$
\end_inset

 be a random variable tha takes only integer values.
 Suppose that 
\begin_inset Formula $X$
\end_inset

 has approximately the normal distribution with mean 
\begin_inset Formula $\mu$
\end_inset

 and variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

.
 Let 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

 be integers, and suppose that we wish to approximate 
\begin_inset Formula $Pr(a\leq X\leq b)$
\end_inset

.
 The correction to the normal distribution approximation for continuity
 is to use 
\begin_inset Formula $\Phi([b+1/2-\mu]/\sigma)-\Phi([a-1/2-\mu]/\sigma)$
\end_inset

 rather than 
\begin_inset Formula $\Phi([b-\mu]/\sigma)-\Phi([a-\mu]/\sigma)$
\end_inset

 as the approximation.
\end_layout

\end_body
\end_document