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Chapter3.lyx
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Chapter3.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
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\end_header
\begin_body
\begin_layout Section*
3.7 Multivariate Distributions
\end_layout
\begin_layout Standard
In this section, we shall extend the result of Bivariate distribution and
marginal distribution for two random variable
\begin_inset Formula $X$
\end_inset
and
\begin_inset Formula $Y$
\end_inset
to an arbitrary finite number
\begin_inset Formula $n$
\end_inset
of random variables
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset
.
In general, the joint distribution of more than two random variables is
called a multivariate distribution.
The theory of statistical inference (Chapter 7) relies on mathematical
models for observable data in which each observation is a random variable.
For this reason, multivariate distributions arise naturally in the mathematical
models for data.
The most commonly used model will be one in which the individual data random
variables are conditionally independent given one or two other random variables.
\end_layout
\begin_layout Subsection*
Histograms
\end_layout
\begin_layout Paragraph
Definition 3.7.9 Histogram
\series medium
.
Let
\begin_inset Formula $x_{1},...,x_{n}$
\end_inset
be a collection of numbers that all lie between two values
\begin_inset Formula $a<b$
\end_inset
.
That is,
\begin_inset Formula $a\leq x_{i}\leq b$
\end_inset
for all
\begin_inset Formula $i=1,...,n$
\end_inset
.
Choose some integer
\begin_inset Formula $k\geq1$
\end_inset
and divide the interval
\begin_inset Formula $[a,b]$
\end_inset
into k equal-length subintervals of length
\begin_inset Formula $(b-a)/k$
\end_inset
.
For each subinterval, count how many of the number
\begin_inset Formula $x_{1},..,x_{n}$
\end_inset
are in the subinterval.
Let
\begin_inset Formula $c_{i}$
\end_inset
be the count for subinterval
\begin_inset Formula $i$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
.
Choose a number
\begin_inset Formula $r>0$
\end_inset
.
(Typically,
\begin_inset Formula $r=1$
\end_inset
or
\begin_inset Formula $r=n$
\end_inset
or
\begin_inset Formula $r=n(b-a)/k$
\end_inset
).
Draw a two-dimensional graph with the horizontal axis running from
\begin_inset Formula $a$
\end_inset
to
\begin_inset Formula $b$
\end_inset
.
For each subinterval
\begin_inset Formula $i=1,...,k$
\end_inset
draw a rectangular bar of width
\begin_inset Formula $(b-a)/k$
\end_inset
and height equal
\begin_inset Formula $c_{i}/r$
\end_inset
over the midpoint of the
\begin_inset Formula $i$
\end_inset
th interval.
Such a graph is called a histogram.
\end_layout
\begin_layout Paragraph
\series medium
The choice of the number
\begin_inset Formula $r$
\end_inset
in the definition of histogram depends on what one wishes to be displayed
on the vertical axis.
The shape of the histogram is identical regardless of what value one chooses
for
\begin_inset Formula $r$
\end_inset
.
With
\begin_inset Formula $r=1$
\end_inset
, he height of each bar is the raw count for each subinterval, and counts
are displayed on the vertical axis.
\series default
This is as normalized value to visualize.
\end_layout
\begin_layout Subsection*
3.8 Functions of a Random Variables
\end_layout
\begin_layout Standard
\series bold
Theorem 3.8.1
\series default
Function of a Discrete Random Variable.
Let
\begin_inset Formula $X$
\end_inset
have a discrete distribution with p.d.
\begin_inset Formula $f$
\end_inset
, and let
\begin_inset Formula $Y=r(X)$
\end_inset
for some function of
\begin_inset Formula $r$
\end_inset
defined on the set of possible values of
\begin_inset Formula $X$
\end_inset
.
For each possible value
\begin_inset Formula $y$
\end_inset
of
\begin_inset Formula $Y$
\end_inset
, the p.f.f
\begin_inset Formula $g$
\end_inset
of
\begin_inset Formula $Y$
\end_inset
is :
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y)=Pr(Y=y)=Pr[r(X)=y]=\sum_{x:r(x)=y}f(x)
\]
\end_inset
\end_layout
\begin_layout Paragraph*
\series medium
Suppose that the p.d.f.
of
\begin_inset Formula $X$
\end_inset
is
\begin_inset Formula $f$
\end_inset
and that another random variable is defined as
\begin_inset Formula $Y=r(X)$
\end_inset
.
For each real number
\begin_inset Formula $y$
\end_inset
, the c.d.f.
\begin_inset Formula $G(y)$
\end_inset
of
\begin_inset Formula $Y$
\end_inset
can be derived as follows:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
G(y)=Pr(Y\leq y)=Pr[r(X)\leq y]=\int_{\{x:r(x)\leq y\}}f(x)dx.
\]
\end_inset
\end_layout
\begin_layout Paragraph*
\series medium
If the random variable
\begin_inset Formula $Y$
\end_inset
also has a continuous distribution, its p.d.f.
\begin_inset Formula $g$
\end_inset
can be obtained from the relation
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y)=\frac{dG(y)}{dy}
\]
\end_inset
\end_layout
\begin_layout Paragraph*
This relation is satisfied at every point
\begin_inset Formula $y$
\end_inset
at which
\begin_inset Formula $G$
\end_inset
is differentiable.
\end_layout
\begin_layout Standard
\series bold
Theorem 3.8.2
\series default
Linear Function.
Suppose
\begin_inset Formula $X$
\end_inset
is a random variable for which the p.d.f.
is
\begin_inset Formula $f$
\end_inset
and that
\begin_inset Formula $Y=aX+b(a\neq0).$
\end_inset
Then the p.d.f.
of
\begin_inset Formula $Y$
\end_inset
is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y)=\frac{1}{|a|}f(\frac{y-b}{a});for-\infty<y<\infty
\]
\end_inset
\end_layout
\begin_layout Standard
and 0 otherwise.
\end_layout
\begin_layout Paragraph*
Theorem 3.8.3
\series medium
Probability Integral Transformation.
Let
\begin_inset Formula $X$
\end_inset
have continuous c.d.f.
\begin_inset Formula $F$
\end_inset
, and let
\begin_inset Formula $Y=F(X)$
\end_inset
.
(This transformation from
\begin_inset Formula $X$
\end_inset
to
\begin_inset Formula $Y$
\end_inset
is called the probability integral transformation.) The distribution of
\begin_inset Formula $Y$
\end_inset
is the uniform distribution on the interval
\begin_inset Formula $[0,1]$
\end_inset
.
\end_layout
\begin_layout Standard
\series bold
Corollary 3.8.1
\series default
Let
\begin_inset Formula $Y$
\end_inset
have the uniform distribution on the interval
\begin_inset Formula $[0,1],$
\end_inset
and let
\begin_inset Formula $F$
\end_inset
be a continuous c.d.f with quantile function
\begin_inset Formula $F^{-1}$
\end_inset
.
The
\begin_inset Formula $X=F^{-1}(Y)$
\end_inset
has c.d.f.
\begin_inset Formula $F$
\end_inset
.
\end_layout
\begin_layout Paragraph*
Theorem 3.8.4
\series medium
Let
\begin_inset Formula $X$
\end_inset
be a random variable for which the p.d.f.
is
\begin_inset Formula $f$
\end_inset
and for which
\begin_inset Formula $Pr(a<X<b)=1$
\end_inset
.
(Here, a and/or b can be either finite or infinite).
Let
\begin_inset Formula $Y=r(X)$
\end_inset
, and suppose that
\begin_inset Formula $r(x)$
\end_inset
is differentiable and one-to-one for
\begin_inset Formula $a<x<b$
\end_inset
.
Let
\begin_inset Formula $(\alpha,\beta)$
\end_inset
be the image of the interval
\begin_inset Formula $(a,b)$
\end_inset
under the function
\begin_inset Formula $r$
\end_inset
.
Let
\begin_inset Formula $s(y)$
\end_inset
be inverse function of
\begin_inset Formula $r(x)$
\end_inset
for
\begin_inset Formula $\alpha<y<\beta.$
\end_inset
The nthe p.d.f.
\begin_inset Formula $g$
\end_inset
of
\begin_inset Formula $Y$
\end_inset
is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(Y)=\begin{cases}
f[s(y)]|\frac{ds(y)}{dy}| & for\alpha<y<\beta\\
0 & otherwise
\end{cases}
\]
\end_inset
\end_layout
\begin_layout Subsection*
3.9 Functions of Two or More Random Variables.
\end_layout
\begin_layout Standard
The general method for solving problems like those of Example 3.9.1 is a straight
forwatd extension of
\series bold
Theorem 3.8.1.
\end_layout
\begin_layout Standard
\series bold
Theorem 3.9.1
\series default
Functions of Discrete Random Variables.
Suppose that
\begin_inset Formula $n$
\end_inset
random variables
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset
have a discrete joint distribution for which the joint p.f.
is
\begin_inset Formula $f$
\end_inset
, and that
\begin_inset Formula $m$
\end_inset
functions
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset
of these
\begin_inset Formula $n$
\end_inset
random variables are defined as follows:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
Y_{1}=r_{1}(X_{1},...,X_{n}),...,Y_{m}=r_{m}(X_{1},...,X_{n})
\]
\end_inset
\end_layout
\begin_layout Standard
For given values
\begin_inset Formula $y_{1},...,y_{m}$
\end_inset
of the
\begin_inset Formula $m$
\end_inset
random variables
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset
, let
\begin_inset Formula $A$
\end_inset
denote the set of all points
\begin_inset Formula $(x_{1},...,x_{n})$
\end_inset
such that
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
r_{1}(x_{1},...,x_{n})=y_{1};...;r_{m}(x_{1},...,x_{n})=y_{m}.
\]
\end_inset
\end_layout
\begin_layout Standard
Then the value of the joint p.f.
\begin_inset Formula $g$
\end_inset
of
\begin_inset Formula $Y_{1},...,Y_{m}$
\end_inset
is specified at the point
\begin_inset Formula $(y_{1},...,y_{m})$
\end_inset
by the relation
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y_{1},...,y_{m})=\sum_{(x_{1},...,x_{n})\in A}f(x_{1},...,x_{n})
\]
\end_inset
\end_layout
\begin_layout Standard
\series bold
Theorem 3.9.2
\series default
Binomial and Bernoulli Distributions Assume that
\begin_inset Formula $X_{1},...,X_{n}$
\end_inset
are i.i.d.
random variables having the Bernoulli distribution with parameter
\begin_inset Formula $p$
\end_inset
.
Let
\begin_inset Formula $Y=X_{1}+...+X_{n}$
\end_inset
.
Then
\begin_inset Formula $Y$
\end_inset
has the binomial distribution with parameters
\begin_inset Formula $n$
\end_inset
and
\begin_inset Formula $p$
\end_inset
.
\end_layout
\begin_layout Standard
\series bold
Question:
\series default
In Binomial case is not equally likely intuition, so explain how it different
with equally likely as
\series bold
Example 3.9.2.
\end_layout
\begin_layout Subsection*
Random Variables with a Continuous Joint Distribution
\end_layout
\begin_layout Standard
Example 3.9.4 is an example of a brute-force method that is always availabel
for finding the distribution of a function of several random variables,
however, it might be difficult to apply in individual cases.
\end_layout
\begin_layout Paragraph*
Theorem 3.9.3
\series medium
Brute-Force Distribution of a Function.
Suppose that the joint p.d.f of
\begin_inset Formula $X=(X_{1},...,X_{n})$
\end_inset
is
\begin_inset Formula $f(\boldsymbol{x})$
\end_inset
and that
\begin_inset Formula $Y=r(\boldsymbol{X}).$
\end_inset
For each real number
\begin_inset Formula $y$
\end_inset
, define
\begin_inset Formula $A_{y}=\{\boldsymbol{x:}r(\boldsymbol{x})\leq y\}.$
\end_inset
Then the c.d.f
\begin_inset Formula $G(y)$
\end_inset
of
\begin_inset Formula $Y$
\end_inset
is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
G(y)=\int\ldots_{A_{y}}\int f(x)dx
\]
\end_inset
\end_layout
\begin_layout Standard
If the distribution of
\begin_inset Formula $Y$
\end_inset
also is continuous, then the p.d.f of
\begin_inset Formula $Y$
\end_inset
can be found by differentiating the c.d.f.
\begin_inset Formula $G(y)$
\end_inset
.
\end_layout
\begin_layout Standard
A popular special case of
\series bold
Theorem 3.9.3
\series default
is the following.
\end_layout
\begin_layout Paragraph*
Theorem 3.9.4 Linear Function of Two Random Variables.
\series medium
Let
\begin_inset Formula $X_{1}$
\end_inset
and
\begin_inset Formula $X_{2}$
\end_inset
have joint p.d.f
\begin_inset Formula $f(x_{1},x_{2})$
\end_inset
, and let
\begin_inset Formula $Y=\alpha_{1}X_{1}+\alpha_{2}X_{2}+b$
\end_inset
with
\begin_inset Formula $a_{1}\neq0$
\end_inset
.
The n
\begin_inset Formula $Y$
\end_inset
has a continous distribution whose p.d.f is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y)=\int_{-\infty}^{\infty}f(\frac{y-b-a_{2}x_{2}}{a_{1}},x_{2})\frac{1}{|a_{1}|}dx_{2}
\]
\end_inset
\end_layout
\begin_layout Paragraph*
Definition 3.9.1 Convolution.
\series medium
Let
\begin_inset Formula $X_{1}$
\end_inset
and
\begin_inset Formula $X_{2}$
\end_inset
be independent continuos random variables and let
\begin_inset Formula $Y=X_{1}+X_{2}$
\end_inset
.
The distribution of
\begin_inset Formula $Y$
\end_inset
is called the convolution of the distributions of
\begin_inset Formula $X_{1}$
\end_inset
and
\begin_inset Formula $X_{2}$
\end_inset
.
The p.d.f of
\begin_inset Formula $Y$
\end_inset
is sometime called the convolution of the p.d.f.'s of
\begin_inset Formula $X_{1}$
\end_inset
and
\begin_inset Formula $X_{2}$
\end_inset
.
\end_layout
\begin_layout Standard
If we let the p.d.f of
\begin_inset Formula $X_{i}$
\end_inset
be
\begin_inset Formula $f_{i}$
\end_inset
for
\begin_inset Formula $i=1,2$
\end_inset
in definition 3.9.1, then Theorem 3.9.4 (with
\begin_inset Formula $a_{1}=a_{2}=1$
\end_inset
and
\begin_inset Formula $b=0$
\end_inset
) says that the p.d.f of
\begin_inset Formula $Y=X_{1}+X_{2}$
\end_inset
is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
g(y)=\int_{-\infty}^{\infty}f_{1}(y-z)f_{2}(z)dz
\]
\end_inset
\end_layout
\begin_layout Paragraph*
\series medium
A popular way to describe how spread out is a random sample is to use the
distance from the minimum to the maximum, which is called the range of
the random sample.
We can combine the result from the end of
\series default
Example 3.9.6
\series medium
with
\series default
Theorem 3.9.4
\series medium
to find the p.d.f of the range.
\end_layout
\begin_layout Standard
\begin_inset Graphics
filename pasted1.png
\end_inset
\end_layout
\begin_layout Paragraph*
\series medium
Next, we state without proof a generalization of
\series default
Theorem 3.8.4
\series medium
to the case of several random variables.
The proof of
\series default
Theorem 3.9.5
\series medium
is based on the differentiable one-to-one transformations (strictly increasing
or decreasing function) in advanced calculus.
\end_layout
\begin_layout Standard
\series bold
Theorem 3.9.5