chapter4_theory.lyx

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\begin_body

\begin_layout Part*
Chapter 4
\end_layout

\begin_layout Section*
4.2 Properties of Expectations 
\end_layout

\begin_layout Standard

\series bold
Theorem 4.2.1 
\series default
Linear Function.
 If 
\begin_inset Formula $Y=aX+b$
\end_inset

, where 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

 are finite constants, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(Y)=aE(X)+b
\]

\end_inset


\end_layout

\begin_layout Paragraph*

\series bold
Theorem 4.2.2 
\series default
If there exists a constant such that 
\begin_inset Formula $Pr(X\ge a)=1$
\end_inset

, then 
\begin_inset Formula $E(X)\geq a$
\end_inset

.
 If there exists a constant 
\begin_inset Formula $b$
\end_inset

 such that 
\begin_inset Formula $Pr(X\le b)=1$
\end_inset

, then 
\begin_inset Formula $E(X)\le b$
\end_inset


\end_layout

\begin_layout Standard
It follows from Theorem 4.2.2 that if 
\begin_inset Formula $Pr(a\le X\leq b)=1$
\end_inset

, then 
\begin_inset Formula $a\leq E(X)\leq b$
\end_inset

.
\end_layout

\begin_layout Paragraph*
Corollary 4.2.1 
\series medium
If 
\begin_inset Formula $X=c$
\end_inset

 with probability 1, then 
\begin_inset Formula $E(X)=c$
\end_inset

.
\end_layout

\begin_layout Paragraph*
Theorem 4.2.3 
\series medium
Suppose that 
\begin_inset Formula $E(X)=a$
\end_inset

 and that either 
\begin_inset Formula $Pr(X\geq a)=1$
\end_inset

 or 
\begin_inset Formula $Pr(X\leq a)=1$
\end_inset

.
 Then 
\begin_inset Formula $Pr(X=a)=1$
\end_inset


\end_layout

\begin_layout Standard

\series bold
Theorem 4.2.4 
\series default
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are 
\begin_inset Formula $n$
\end_inset

 random variables such that each expectation 
\begin_inset Formula $E(X_{i})$
\end_inset

 is finite 
\begin_inset Formula $(i=1,...,n)$
\end_inset

, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(X_{1}+...+X_{n})=E(X_{1})+...+E(X_{n})
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Corollary 4.2.2 
\series medium
Assume that 
\begin_inset Formula $E(X_{i})$
\end_inset

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

.
 For all constants 
\begin_inset Formula $a_{1},...,a_{n}$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(a_{1}X_{1}+...+a_{n}X_{n}+b)=a_{1}E(X_{1})+...+a_{n}E(X_{n})+b
\]

\end_inset


\end_layout

\begin_layout Paragraph*

\series medium
Note: In General, 
\begin_inset Formula $E[g(X)]\neq g(E(X)).$
\end_inset

 Theorems 4.2.1 and 4.2.4 imply that if 
\begin_inset Formula $g$
\end_inset

 is a linear function of a random vector 
\begin_inset Formula $X$
\end_inset

, then 
\begin_inset Formula $E[g(X)]=g(E(X))$
\end_inset

.
 For a nonlinear function 
\begin_inset Formula $g$
\end_inset

, Jensen's inequality (Theorem 4.2.5) gives a relationship between 
\begin_inset Formula $E[g(X)]$
\end_inset

 and 
\begin_inset Formula $g(E(X))$
\end_inset

 for another special class of functions.
\end_layout

\begin_layout Paragraph*
Definition 4.2.1 
\series medium
Convex Functions.
 A function 
\begin_inset Formula $g$
\end_inset

 of a vector argument is convex if, for every 
\begin_inset Formula $\alpha\in(0,1)$
\end_inset

, and every 
\series default
x 
\series medium
and 
\series default
y
\series medium
,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
g[\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}]\geq\alpha g(\boldsymbol{x})+(1-\alpha)g(\boldsymbol{y}).
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Theorem 4.2.5 
\series medium
Jensen's inequality.
 Let 
\begin_inset Formula $g$
\end_inset

 be a convex function, and let 
\series default
X 
\series medium
be a random vector with finite mean.
 Then 
\begin_inset Formula $E[g(X)]\geq g(E(X))$
\end_inset

.
\end_layout

\begin_layout Standard
Provement of special case in Exercise 13
\end_layout

\begin_layout Subsection*
Expectation of a Product of Independent Random Variables
\end_layout

\begin_layout Standard

\series bold
Theorem 4.2.6.
 
\series default
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are 
\begin_inset Formula $n$
\end_inset

 independent random variables such that each expectation 
\begin_inset Formula $E(X_{i})$
\end_inset

 is finite (
\begin_inset Formula $i=1,...,n$
\end_inset

), then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(\prod_{i=1}^{n}X_{i})=\prod_{i=1}^{n}E(X_{i})
\]

\end_inset


\end_layout

\begin_layout Standard
The difference between Theorem 4.2.4 and Theorem 4.2.6 should be emphasized.
 Expectation of the sum of a group of random variables is always equal to
 the sum of their individual expectations.
 However, the expectation of the product of a group of random variables
 is 
\series bold
not always 
\series default
equal to the product of their individual expectations.
 If the random variables are independent, then this equality will also hold.
\end_layout

\begin_layout Subsection*
Expectation for Nonnegative Distributions.
\end_layout

\begin_layout Standard

\series bold
Theorem 4.2.7 
\series default
Integer-Valued Random Variables.
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable that can take only the values 
\begin_inset Formula $0,1,2,...$
\end_inset

 Then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(X)=\sum_{n=1}^{\infty}Pr(X\ge n).(4.2.7)
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Theorem 4.2.8 
\series medium
General Nonnegative Random Variable.
 Let 
\begin_inset Formula $X$
\end_inset

 be a nonnegative random variable with c.d.f 
\begin_inset Formula $F$
\end_inset

 .
 Then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E(X)=\int_{0}^{\infty}[1-F(x)]dx.(4.2.9)
\]

\end_inset


\end_layout

\begin_layout Standard
The proof of Theorem 4.2.8 is left in Exercises 1 and 2 in Sec.
 4.9.
\end_layout

\begin_layout Paragraph*
Example 4.2.9 is good example applying Theorem 4.2.7 
\series medium
instead of using computing expectation in normal way.
\end_layout

\begin_layout Paragraph*
Summary 
\series medium
The mean of a linear function of a random vector is the linear function
 of the mean.
 In particular, the mean of a sum is the sum of the means.
 As an example, the mean of the binomial distribution with parameters n
 and p is 
\begin_inset Formula $np$
\end_inset

.
 No such relationship holds in general for nonlinear functions.
 For independent random variables, the mean of product is the product of
 the means.
\end_layout

\begin_layout Section*
4.3 Variance 
\end_layout

\begin_layout Standard
....
 The variance also plays an important role in the approximation methods
 that arise in Chapter 6.
\end_layout

\begin_layout Paragraph*
Definition 4.3.1 
\series medium
Variance/Standard Deivation.
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable with finite mean 
\begin_inset Formula $\mu=E(X)$
\end_inset

.
 The variance of 
\begin_inset Formula $X$
\end_inset

, denoted by 
\begin_inset Formula $Var(X)$
\end_inset

, is defined as follows:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(X)=E[(X-\mu)^{2}].(4.3.1)
\]

\end_inset


\end_layout

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

 has infinite mean or if the mean of 
\begin_inset Formula $X$
\end_inset

 does not exist, we say that 
\begin_inset Formula $Var(X)$
\end_inset

 does not exist.
\end_layout

\begin_layout Standard
If the expectation in Eq.
 (4.3.1) is finite, we say that 
\begin_inset Formula $Var(X)$
\end_inset

 and the standard deviation of 
\begin_inset Formula $X$
\end_inset

 are infinite.
 
\end_layout

\begin_layout Paragraph*
Theorem 4.3.1 
\series medium
Alternative Method for Calculating the Variance.
 For every random variable 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $Var(X)=E(X^{2})-[E(X)]^{2}$
\end_inset

.
\end_layout

\begin_layout Standard
The variance of a distribution, as well as the mean can be made arbitrarily
 large by placing even a very small but positive amount of probability far
 enough from the origin on the real line (Se 
\series bold
Example 4.3.5 
\series default
for detail).
\end_layout

\begin_layout Subsection*
Properties of the Variance 
\end_layout

\begin_layout Standard

\series bold
Theorem 4.3.2 
\series default
For each 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $Var(X)\geq0$
\end_inset

.
 If 
\begin_inset Formula $X$
\end_inset

 is a bounded random variable, then 
\begin_inset Formula $Var(X)$
\end_inset

 must exist and be finite.
\end_layout

\begin_layout Paragraph*
Theorem 4.3.3 
\series medium

\begin_inset Formula $Var(X)=0$
\end_inset

 if and only if there exists a constant c such that 
\begin_inset Formula $P(X=c)=1$
\end_inset


\end_layout

\begin_layout Standard

\series bold
Theorem 4.3.4 
\series default
For constants 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

, let 
\begin_inset Formula $Y=aX+b$
\end_inset

.
 Then 
\begin_inset Formula $Var(Y)=a^{2}Var(X),$
\end_inset

 and 
\begin_inset Formula $\sigma_{Y}=|a|\sigma_{y}$
\end_inset


\end_layout

\begin_layout Standard
It follows from Theorem 4.3.4 that 
\begin_inset Formula $Var(-X)=Var(X)$
\end_inset


\end_layout

\begin_layout Paragraph*
Theorem 4.3.5 
\series medium
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are independent random variables with finite means, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(X_{1}+...+X_{n})=Var(X_{1})+...+Var(X_{n})
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Corollary 4.3.1 
\series medium
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are independent random variables with finite means, and if 
\begin_inset Formula $a_{1},...,a_{n}$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

 are arbitrary constant, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(a_{1}X_{1}+...+a_{n}X_{n}+b)=a_{1}^{2}Var(X_{1})+...+a_{n}^{2}Var(X_{n}).
\]

\end_inset


\end_layout

\begin_layout Paragraph*
In Example 4.3.7 about portfolio 
\series medium
One method for comparing a class of portfolios is to say that portfolio
 
\begin_inset Formula $A$
\end_inset

 is at least as good as portfolio 
\begin_inset Formula $B$
\end_inset

 if the mean return for 
\begin_inset Formula $A$
\end_inset

 is at least as large as the mean return for 
\begin_inset Formula $A$
\end_inset

 is at least as large as the mean return for 
\begin_inset Formula $B$
\end_inset

 and if the variance for 
\begin_inset Formula $A$
\end_inset

 is 
\series default
no larger than the variance of B.
 (See Markowitz, 1987, for a classic treatmean of such methods).

\series medium
 The reason for preferring smaller variance is that large variance is associated
 with large deviations from the mean, and for portfolios with common mean,
 some of the large deviations are going to be below the mean, leading to
 the rish of large losses.
 Figure 4.7 is a plot of the pairs (mean, variance) for all of the possible
 portfolios in this example.
 That is, for each (
\begin_inset Formula $s_{1},s_{2},s_{3}$
\end_inset

) that satisfy (4.3.2): 
\begin_inset Formula $60s_{1}+48s_{2}+s_{3}=100000$
\end_inset

, there is a point in the outlined region of Fig.
 4.7.
 The points to the right and toward the bottom are those that have the largest
 mean return for a fixed variance, and the ones that have the smallest variance
 for a fixed mean return.
 These portfolios are called efficient.
\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename portfolio_comparision.png

\end_inset


\end_layout

\begin_layout Subsection*
Interquatile Range 
\end_layout

\begin_layout Standard
There is a measure of spread that exists for every distribution, regardless
 of whether or not the distribution has a mean or variance.
 
\end_layout

\begin_layout Paragraph*
Definition 4.3.2 
\series medium
Interquartile Range (IQR).
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable with quantile function 
\begin_inset Formula $F^{-1}(p)$
\end_inset

 for 
\begin_inset Formula $0<p<1$
\end_inset

.
 The 
\emph on
interquartile range 
\emph default
(IQR) is defined to be 
\begin_inset Formula $F^{-1}(0.75)-F^{-1}(0.25)$
\end_inset

.
 
\end_layout

\begin_layout Standard
In words, the IQR is the length of the interval that contains the middle
 half of the distribution.
\end_layout

\begin_layout Standard
IQR is 
\series bold
a measure of spread that exists for every distribution.
\end_layout

\begin_layout Section*
4.4 Moments
\end_layout

\begin_layout Standard

\emph on
The moment generating function is a related tool that aids in deriving distribut
ions of sums of independent random variables and limiting properties of
 distributions.
\end_layout

\begin_layout Standard

\series bold
Existence of Moments
\end_layout

\begin_layout Standard
\begin_inset Formula $E[X^{k}]$
\end_inset

 is called the kth moment of 
\begin_inset Formula $X$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $k$
\end_inset

th moment exists if and only if 
\begin_inset Formula $E[|X|^{k}]<\infty$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $P[a\leq X\le b]=1$
\end_inset

 (
\begin_inset Formula $X$
\end_inset

 is bounded) then all moments of 
\begin_inset Formula $X$
\end_inset

 must necessarily exists.
 It is possible that all moments of 
\begin_inset Formula $X$
\end_inset

 exists even though 
\begin_inset Formula $X$
\end_inset

 is not bounded.
 
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.1 , 
\series default
If 
\begin_inset Formula $E[|X|^{k}]<\infty$
\end_inset

 for some positive integer 
\begin_inset Formula $k$
\end_inset

 then 
\begin_inset Formula $E[|X|^{j}]<\infty$
\end_inset

 for every positive integer 
\begin_inset Formula $j$
\end_inset

 such that 
\begin_inset Formula $j<k$
\end_inset


\end_layout

\begin_layout Standard

\series bold
Central Moments 
\series default
Suppose that 
\begin_inset Formula $X$
\end_inset

 is a random variable for which 
\begin_inset Formula $E(X)=\mu$
\end_inset

 .
 For every positive integer 
\begin_inset Formula $k$
\end_inset

, the expectation 
\begin_inset Formula $E[(X-\mu)^{k}]$
\end_inset

 is called the 
\begin_inset Formula $k$
\end_inset

th 
\emph on
central moment 
\emph default
of 
\begin_inset Formula $X$
\end_inset

 or 
\begin_inset Formula $k$
\end_inset

th 
\emph on
moment of 
\begin_inset Formula $X$
\end_inset

 about the mean.
 
\emph default
In particular, in accordance with this terminology, the variance of 
\begin_inset Formula $X$
\end_inset

 is the second central moment of 
\begin_inset Formula $X$
\end_inset

.
 
\end_layout

\begin_layout Standard

\series bold
Definition 4.4.1 
\series default
Skewness (measurement of symmetry) .
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable with mean 
\begin_inset Formula $\mu$
\end_inset

, standard deviation 
\begin_inset Formula $\sigma$
\end_inset

 and finite third moment.
 The 
\emph on
skewness 
\emph default
of 
\begin_inset Formula $X$
\end_inset

 is defined to be 
\begin_inset Formula $E[(X-\mu)^{3}]/\sigma^{3}$
\end_inset

.
\end_layout

\begin_layout Standard
Dividing 
\begin_inset Formula $\sigma^{3}$
\end_inset

 is to make the skewness measure only the lack of symmetry rather than the
 spread of the distribution.
 
\end_layout

\begin_layout Standard

\series bold
Moment Generating Functions
\end_layout

\begin_layout Standard

\series bold
Definition 4.4.2 
\series default
Moment Generating Function.
 Let 
\begin_inset Formula $X$
\end_inset

 be a random variable.
 For each real number 
\begin_inset Formula $t$
\end_inset

 , define: 
\begin_inset Formula $\psi(t)=E(e^{tX})$
\end_inset

.
\end_layout

\begin_layout Standard
The function 
\begin_inset Formula $\psi(t)$
\end_inset

 is called the 
\emph on
moment generating function
\emph default
 (m.g.f) of 
\begin_inset Formula $X$
\end_inset

.
 
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.2 
\series default
Let 
\begin_inset Formula $X$
\end_inset

 be a random variables whose m.g.f.
 
\begin_inset Formula $\psi(t)$
\end_inset

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

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

.
 Then, for each integer 
\begin_inset Formula $n>0$
\end_inset

, the nth moment of 
\begin_inset Formula $X$
\end_inset

 , 
\begin_inset Formula $E(X^{n})$
\end_inset

, is finite and equals the 
\begin_inset Formula $n$
\end_inset

th derivative 
\begin_inset Formula $\psi^{(n)}(t)$
\end_inset

 at 
\begin_inset Formula $t=0$
\end_inset

.
 That is, 
\begin_inset Formula $E[X^{n}]=\psi^{(n)}(0)$
\end_inset

 for 
\begin_inset Formula $n=1,2,3...$
\end_inset

 .
\end_layout

\begin_layout Standard

\series bold
Properties of Moment Generating Functions
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.3, 
\series default
Let 
\begin_inset Formula $X$
\end_inset

 be a random variable for which the m.g.f is 
\begin_inset Formula $\psi_{1}$
\end_inset

; let 
\begin_inset Formula $Y=aX+b$
\end_inset

, where 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

 are given constants; and let 
\begin_inset Formula $\psi_{2}$
\end_inset

denote the m.g.f of 
\begin_inset Formula $Y$
\end_inset

 .
 Then for every value of 
\begin_inset Formula $t$
\end_inset

 such that 
\begin_inset Formula $\psi_{1}(at)$
\end_inset

 is finite: 
\begin_inset Formula $\psi_{2}(t)=e^{bt}\psi_{1}(at)$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.4 
\series default
Suppose that 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are n independent random variables; and for 
\begin_inset Formula $i=1,...,n$
\end_inset

.
 let 
\begin_inset Formula $\psi_{i}$
\end_inset

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

.
 Let 
\begin_inset Formula $Y=X_{1}+...+X_{n}$
\end_inset

, ad let m.g.f.
 of 
\begin_inset Formula $Y$
\end_inset

 be denote by 
\begin_inset Formula $\psi$
\end_inset

.
 Then for every value of 
\begin_inset Formula $t$
\end_inset

 such that 
\begin_inset Formula $\psi_{i}(t)$
\end_inset

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

 : 
\begin_inset Formula $\psi(t)=\prod_{i=1}^{n}\psi_{i}(t)$
\end_inset


\end_layout

\begin_layout Standard

\series bold
Uniqueness of Moment Generating Functions .
 
\series default
We shall now state one more important property of the m.g.f.
 
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.5.
 
\series default
If the m.g.f.'s of two random variables 
\begin_inset Formula $X_{1}$
\end_inset

and 
\begin_inset Formula $X_{2}$
\end_inset

are finite and identical for all values of 
\begin_inset Formula $t$
\end_inset

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

, then the probability distribution of 
\begin_inset Formula $X_{1}$
\end_inset

and 
\begin_inset Formula $X_{2}$
\end_inset

 must be identical .
\end_layout

\begin_layout Standard

\series bold
Why: 
\series default
Dont understand this theorem ??
\end_layout

\begin_layout Standard
m.g.f.
 is another way to characterize the distribution of a random variable.
\end_layout

\begin_layout Standard

\series bold
The Additive Property of the Binomial Distribution 
\series default
Moment generating functions provide a simple way to derive the distribution
 of the sum of two independent binomial random variables with the same second
 parameter.
\end_layout

\begin_layout Standard

\series bold
Theorem 4.4.6 
\series default
If 
\begin_inset Formula $X_{1}$
\end_inset

and 
\begin_inset Formula $X_{2}$
\end_inset

 are independent random variables, and if 
\begin_inset Formula $X_{i}$
\end_inset

 has the binomial distribution with parameters 
\begin_inset Formula $n_{i}$
\end_inset

 and 
\begin_inset Formula $p(i=1,2)$
\end_inset

, then 
\begin_inset Formula $X_{1}+X_{2}$
\end_inset

has the binomial distribution with parameters 
\begin_inset Formula $n_{1}+n_{2}$
\end_inset

 and 
\begin_inset Formula $p$
\end_inset

.
\end_layout

\begin_layout Subsection*
4.6 Covariance and Correlation 
\end_layout

\begin_layout Standard
When we are interested in the joint distribution of two random variables,
 it is useful to have a summary of how much the two random variables depend
 on each other.
 The covariance and correlation are attempts to measure that dependence,
 but they only capture a particular type of dependence, namely linear dependence.
\end_layout

\begin_layout Subsubsection*
Covariance 
\end_layout

\begin_layout Standard
When we consider the joint distribution of two random variables, the means,
 the medians, and the variances of the variables provide useful information
 about their marginal distributions.
 However, these values do not provide any information about the relationship
 between the two variables or about their tendency to vary together rather
 than independently.
 In this section, we shall introduce summaries of a joint distribution that
 enable us to measure the association between two random variables, determine
 the variance of the sum of an arbitrary number of dependent random variables
 and 
\series bold
predict the value of one random variable by using the observed value of
 some other related variable.
\end_layout

\begin_layout Paragraph*
Definition 4.6.1.
 
\series medium
Covariance.
 Let 
\begin_inset Formula $X$
\end_inset

and 
\begin_inset Formula $Y$
\end_inset

 be random variabeles having finite means.
 Let 
\begin_inset Formula $E(X)=\mu_{X}$
\end_inset

 and 
\begin_inset Formula $E(Y)=\mu_{Y}$
\end_inset

 .
 The 
\emph on
covariance of 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 , 
\emph default
is defined as 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Cov(X,Y)=E[(X-\mu_{X})(Y-\mu_{Y})]
\]

\end_inset

 (4.6.1)
\end_layout

\begin_layout Standard
It can be shown (Exercise 2) that if both 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 have finite variance, then the expectation in Eq.
 (4.6.1) will exists and 
\begin_inset Formula $Cov(X,Y)$
\end_inset

will be finite .
 However, the value of 
\begin_inset Formula $Cov(X,Y)$
\end_inset

 can be positive , negative, or zero.
\end_layout

\begin_layout Paragraph*
Theorem 4.6.1 
\series medium
For all random variables 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 such that 
\begin_inset Formula $\sigma_{X}^{2}<\infty$
\end_inset

 and 
\begin_inset Formula $\sigma_{Y}^{2}<\infty$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Cov(X,Y)=E[XY]-E[X]E[Y]
\]

\end_inset


\end_layout

\begin_layout Paragraph*

\series medium
The covariance between 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 is intended to measure the degree to which 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 tend to be large at the same time or the degree to which one tends to be
 large while the other is small.
 Some intution about this interpretation can be gathered from a care look
 at Eq.
 (4.6.1) .
 For example, suppose that 
\begin_inset Formula $Cov(X,Y)$
\end_inset

 is positive.
 Then 
\begin_inset Formula $X>\mu_{X}$
\end_inset

 and 
\begin_inset Formula $Y>\mu_{Y}$
\end_inset

 must occur together and/or 
\begin_inset Formula $X<\mu_{X}$
\end_inset

 and 
\begin_inset Formula $Y<\mu_{Y}$
\end_inset

 must occur together to a larger extent than 
\begin_inset Formula $X<\mu_{X}$
\end_inset

occurs with 
\begin_inset Formula $Y>\mu_{Y}$
\end_inset

and 
\begin_inset Formula $X>\mu_{X}$
\end_inset

occurs with 
\begin_inset Formula $Y<\mu_{Y}$
\end_inset

.
 Otherwise , the mean would be negative.
 Similar for 
\begin_inset Formula $Cov(X,Y)$
\end_inset

be negative.
 If 
\begin_inset Formula $Cov(X,Y)=0$
\end_inset

, then the extent to which 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are on the same sides of their respective means exactly balances the extent
 to which they are on opposite sides of their means.
\end_layout

\begin_layout Subsubsection*
Correlation 
\end_layout

\begin_layout Standard
Although 
\begin_inset Formula $Cov(X,Y)$
\end_inset

 gives a numerical measure of the degree to which 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

vary together, the magnitude of 
\begin_inset Formula $Cov(X,y)$
\end_inset

 is also influenced by the overall magnitudes of 
\begin_inset Formula $X,Y$
\end_inset

.
 For example, in Exercise 5 in this section, you van prove that 
\begin_inset Formula $Cov(2X,Y)=2Cov(X,y)$
\end_inset

.
 In order to obtain a measure of association between 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 that is 
\emph on
not driven by arbitrary changes in the scales 
\emph default
of one or the other random variable, we define a slightly different quantity
 next.
 
\end_layout

\begin_layout Paragraph*
Definition 4.6.2 
\series medium
Correlation.
 Let 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 be random variables with finite variances 
\begin_inset Formula $\sigma_{X}^{2}$
\end_inset

 and 
\begin_inset Formula $\sigma_{Y}^{2}$
\end_inset

, respectively.
 The the correlation of 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

, which is denoted by 
\begin_inset Formula $\rho(X,Y)$
\end_inset

, is defined as follows: 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\rho(X,Y)=\frac{Cov(X,Y)}{\sigma_{X}\sigma_{Y}}
\]

\end_inset


\end_layout

\begin_layout Standard
In order to deter mine the range of possible values of the correlation 
\begin_inset Formula $\rho(X,Y)$
\end_inset

, we shall need the following result 
\end_layout

\begin_layout Paragraph*
Theorem 4.6.2 
\series medium
Schwarz Inequality.
 For all random variables 
\begin_inset Formula $U$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

 such that 
\begin_inset Formula $E[UV]$
\end_inset

 exists, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
[E(UV)]^{2}\le E(U^{2})E(V^{2})
\]

\end_inset

 (4.6.4)
\end_layout

\begin_layout Paragraph*
Theorem 4.6.3 
\series medium
Cauchy-Schwarz Inequality.
 Let 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 be random variables with finite variance.
 Then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
[Conv(X,Y)]^{2}\le\sigma_{X}^{2}\sigma_{Y}^{2},
\]

\end_inset


\end_layout

\begin_layout Standard
and 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
-1\le\rho(X,Y)\le1
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Definition 4.6.3 
\series medium
Positively/Negatively Correlated/Uncorrelated.
 It is said that 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are positively correlated if 
\begin_inset Formula $\rho(X,Y)>0$
\end_inset

, that 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are negatively correlated if 
\begin_inset Formula $\rho(X,Y)<0$
\end_inset

.
 And that 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are uncorrelated if 
\begin_inset Formula $\rho(X,Y)=0$
\end_inset

.
\end_layout

\begin_layout Standard
It can be seen from Eq.
 (4.6.3) that 
\begin_inset Formula $Cov(X,Y)$
\end_inset

 and 
\begin_inset Formula $\rho(X,Y)$
\end_inset

 must have the same sign, that is, both are positive, ot both are negative,
 or both are zero.
\end_layout

\begin_layout Subsubsection*
Properties of Covariance and Correlation 
\end_layout

\begin_layout Standard
Now present four theorems pertainint to the basic properties of covariance
 and correlation.
 The first theorem shows that independent random variables must be uncorrelated.
\end_layout

\begin_layout Paragraph*
Theorem 4.6.4 
\series medium
If 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are independent random variables with 
\begin_inset Formula $0<\sigma_{X}^{2}<\infty$
\end_inset

 and 
\begin_inset Formula $0<\sigma_{Y}^{2}<\infty$
\end_inset

, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Cov(X,Y)=\rho(X,Y)=0
\]

\end_inset


\end_layout

\begin_layout Paragraph*

\series medium
The converse of Theorem 4.6.4 is not true as a general rule.
 Two dependent random variables can be uncorrelated.
 Indeed, even though 
\begin_inset Formula $Y$
\end_inset

 is an explicit function of 
\begin_inset Formula $X$
\end_inset

, it is possible 
\begin_inset Formula $\rho(X,Y)=0$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Theorem 4.6.5 
\series default
Suppose that 
\begin_inset Formula $X$
\end_inset

 is a random variable such that 
\begin_inset Formula $0<\sigma_{X}^{2}<\infty$
\end_inset

, and 
\begin_inset Formula $Y=aX+b$
\end_inset

 for some constants 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b$
\end_inset

, where 
\begin_inset Formula $a\neq0.$
\end_inset

If 
\begin_inset Formula $a>0$
\end_inset

, then 
\begin_inset Formula $\rho(X,Y)=1$
\end_inset

.
 If 
\begin_inset Formula $a<0$
\end_inset

, then 
\begin_inset Formula $p(X,Y)=-1$
\end_inset


\end_layout

\begin_layout Standard
There is a converse to 
\series bold
Theorem 4.6.5.
 
\series default
That is, 
\begin_inset Formula $|\rho(X,Y)|=1$
\end_inset

 implies that 
\begin_inset Formula $X$
\end_inset

and 
\begin_inset Formula $Y$
\end_inset

 are linearly related (See Exercise 17).
 In general, the value of 
\begin_inset Formula $\rho(X,Y)$
\end_inset

 provides a measure of the extent to which two random variables 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are linearly related.
 If the joint distribution of 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 is relatively concentrated around straight line in the xy-plane that has
 postivie slope, then 
\begin_inset Formula $\rho(X,Y)$
\end_inset

 close to 1.
 Similar if concentraing aroud straight line in xy-plane has negative slope,
 
\begin_inset Formula $\rho(X,Y)$
\end_inset

 close -1.
 We shall consider them again when the bivariate normal distribution is
 introduced and studied in Sec 5.10.
\end_layout

\begin_layout Standard

\series bold
Note: Correlation measures Only Linear Relationship.
 
\series default
A large value of 
\begin_inset Formula $|\rho(X,Y)|$
\end_inset

 means that 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are close to being linearly related and hence are closely related.
 But a small value of 
\begin_inset Formula $|\rho(X,Y)|$
\end_inset

 does not mean that 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are not close to being related.
 Indeed, Examlple 4.6.4 illustrates random variables that are functionally
 related but have 0 correlation.
\end_layout

\begin_layout Paragraph*
Theorem 4.6.6 
\series medium
If 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are random variables such that 
\begin_inset Formula $Var(X)<\infty$
\end_inset

 anf 
\begin_inset Formula $Var(Y)<\infty$
\end_inset

, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Corollary 4.6.1 
\series medium
Let a,b, and c be constants.
 Under the conditions of Theorem 4.6.6, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(aX+bY+c)=a^{2}Var(X)+b^{2}Var(Y)+2abCov(X,Y)
\]

\end_inset


\end_layout

\begin_layout Standard
A particularly useful special case of Corollary 4.6.1 is 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Theorem 4.6.7 
\series medium
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are random variables such that 
\begin_inset Formula $Var(X_{i})<\infty$
\end_inset

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

, then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(\sum_{i=1}^{n}X_{i})=\sum_{i=1}^{n}Var(X_{i})+2\sum\sum_{i<j}Cov(X_{i},X_{j}).
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Crollary 4.6.2 
\series medium
If 
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset

 are uncorrelated random variabltes (that is, if 
\begin_inset Formula $X_{i}$
\end_inset

 and 
\begin_inset Formula $X_{j}$
\end_inset

 are uncorrelated whenever 
\begin_inset Formula $i\neq j$
\end_inset

), then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Var(\sum_{i=1}^{n}X_{i})=\sum_{i=1}^{n}Var(X_{i})
\]

\end_inset


\end_layout

\begin_layout Paragraph*
Note: In General, Variances Add Only for Uncorrelated Random Variabels.
 
\series medium
The variance of a sum of random variables should be calculated using Theorem
 4.6.7 in general.
 Corollary 4.6.2 applies only for uncorrelated random variabels.
\end_layout

\begin_layout Standard

\series bold
Intuition from Exercise
\end_layout

\begin_layout Paragraph*
From exercise 5: 
\begin_inset Formula $Cov(aX+b,cY+d)=acCov(X,Y)$
\end_inset


\end_layout

\begin_layout Standard

\series bold
From exercise 8: 
\begin_inset Formula $Cov(\sum_{i=1}^{m}a_{i}X_{i},\sum_{j=1}^{n}b_{j}Y_{j})=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{i}b_{j}Cov(X_{i},Y_{j})$
\end_inset

.
 
\series default
Tinh chat ket hop
\end_layout

\begin_layout Section*
4.7 Conditional Expectation 
\end_layout

\begin_layout Paragraph*
Theorem 4.7.1 
\series medium
Law of Total Probability for Expectations.
 Let 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 be random variables such that 
\begin_inset Formula $Y$
\end_inset

 has finite mean.
 Then 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E[E(Y|X)]=E(Y)
\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Theorem 4.7.3 
\series default
The prediction 
\begin_inset Formula $d(X)$
\end_inset

 that minimize 
\begin_inset Formula $E\{[Y-d(X)]^{2}\}$
\end_inset

 is 
\begin_inset Formula $d(X)=E[Y|X]$
\end_inset

.
\end_layout

\begin_layout Paragraph*
Theorem 4.7.4 
\series medium
Law of Total Probability for Variances.
 If 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $Y$
\end_inset

 are arbitrary random vairables for which the necessary expectations and
 variances exist, then 
\begin_inset Formula $Var(Y)=E[Var(Y|X)]+Var[E(Y|X)]$
\end_inset


\end_layout

\end_body
\end_document