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\title{MATH/STAT 355: Statistical Theory}
\author{Taylor Okonek}
\date{2023-12-07}
\begin{document}
\maketitle
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\bookmarksetup{startatroot}
\chapter*{Welcome to Statistical
Theory!}\label{welcome-to-statistical-theory}
\addcontentsline{toc}{chapter}{Welcome to Statistical Theory!}
\markboth{Welcome to Statistical Theory!}{Welcome to Statistical
Theory!}
This book contains the course notes for \emph{MATH/STAT 355:}
\emph{Statistical Theory*} at Macalester College, as taught by
Prof.~\href{https://taylorokonek.github.io/}{Taylor Okonek}. These notes
draw from reading guides created by
Prof.~\href{https://kegrinde.github.io/}{Kelsey Grinde}, a little bit
from the textbook, \emph{An Introduction to Mathematical Statistics and
Its Applications} by Richard Larsen and Morris Marx (6th Edition), and
the STAT 512/513 Course Notes developed by Dr.~Michael Perlman at the
University of Washington. As of Spring 2025, this course no longer
requires a textbook, and relies heavily on these course notes instead.
Each chapter of the course notes will contain (at a minimum):
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
Topic Introduction
\item
Learning Objectives
\item
Concept Questions
\item
Definitions
\item
Theorems
\item
Worked Examples
\end{enumerate}
\textbf{I will be editing and adding to these notes throughout Spring
2025, so please check consistently for updates!}
If you find any typos or have other questions, please email
\href{mailto:tokonek@macalester.edu}{\nolinkurl{tokonek@macalester.edu}}.
* \emph{MATH/STAT 355: Statistical Theory} went under the title
\emph{MATH/STAT 455: Mathematical Statistics} prior to Spring 2025. The
course content is largely similar, the differences primarily being the
structure of the course and not the content itself.
\bookmarksetup{startatroot}
\chapter{Probability: A Brief Review}\label{probability-a-brief-review}
\emph{MATH/STAT 355} builds directly on topics covered in
\emph{MATH/STAT 354: Probability}. You're not expected to perfectly
remember everything from \emph{Probability}, but you will need to have
sufficient facility with the following topics covered in this review
Chapter in order to grasp the majority of concepts covered in
\emph{MATH/STAT 355}.
\section{Learning Objectives}\label{learning-objectives}
By the end of this chapter, you should be able to\ldots{}
\begin{itemize}
\item
Distinguish between important probability models (e.g., Normal,
Binomial)
\item
Derive the expectation and variance of a single random variable or a
sum of random variables
\item
Define the moment generating function and use it to find moments or
identify pdfs
\item
Derive pdfs of transformations of continuous random variables
\end{itemize}
\section{Concept Questions}\label{concept-questions}
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
Which probability distributions are appropriate for
\emph{quantitative} (continuous) random variables?
\item
Which probability distributions are appropriate for \emph{categorical}
random variables?
\item
\emph{Independently and Identically Distributed (iid)} random
variables are an incredibly important assumption involved in many
statistical methods. Why do you think it might be important/useful for
random variables to have this property?
\item
Why might we want to be able to derive distribution functions for
transformations of random variables? In what scenarios can you imagine
this being useful?
\end{enumerate}
\section{Definitions}\label{definitions}
You are expected to know the following definitions:
\textbf{Random Variable}
A random variable is a function that takes inputs from a sample space of
all possible outcomes, and outputs real values or probabilities. As an
example, consider a coin flip. The sample space of all possible outcomes
consists of ``heads'' and ``tails'', and each outcome is associated with
a probability (50\% each, for a fair coin). For our purposes, you should
know that random variables have probability density (or mass) functions,
and are either discrete or continuous based on the number of possible
outcomes a random variable may take. Random variables are often denoted
with capital Roman letters, like \(X\), \(Y\), \(Z\), etc.
\textbf{Probability density function} (discrete, continuous)
\begin{itemize}
\tightlist
\item
Note: I don't care if you call a pmf a pdf\ldots{} I will probably do
this continuously throughout the semester. We don't need to be picky
about this in \emph{MATH/STAT 355}.
\end{itemize}
There are many different accepted ways to write the notation for a pdf
of a random variables. Any of the following are perfectly appropriate
for this class: \(f(x)\), \(\pi(x)\), \(p(x)\), \(f_X(x)\). I typically
use either \(\pi\) or \(p\), but might mix it up occasionally.
Key things I want you to know about probability density functions:
\begin{itemize}
\item
\(\pi(x) \geq 0\), everywhere. This should make sense (hopefully)
because probabilities cannot be negative!
\item
\(\int_{-\infty}^\infty \pi(x) = 1\). This should also (hopefully)
makes sense. Probabilities can't be \emph{greater} than one, and the
probability of event occurring \emph{at all (ever)} should be equal to
one, if the event \(x\) is a random variable.
\end{itemize}
\textbf{Cumulative distribution function} (discrete, continuous)
Cumulative distribution functions we'll typically write as \(F_X(x)\).
or \(F(x)\), for short. It is important to know that
\[
F_X(x) = \Pr(X \leq x),
\]
or in words, ``the cumulative distribution function is the probability
that a random variable lies before \(x\).'' If you write \(\Pr(X < x)\)
instead of \(\leq\), you're fine. The probability that a random variable
is exactly one number (for an RV with a continuous pdf) is zero anyway,
so these are the same thing. Key things I want you to know about
cumulative distribution functions:
\begin{itemize}
\item
\(F(x)\) is non-decreasing. This is in part where the ``cumulative''
piece comes in to play. Recall that probabilities are basically
integrals or sums. If we're integrating over something positive, and
our upper bound for our integral \emph{increases}, the area under the
curve (cumulative probability) will increase as well.
\item
\(0 \leq F(x) \leq 1\) (since probabilities have to be between zero
and one!)
\item
\(\Pr(a < X \leq b) = F(b) - F(a)\) (because algebra)
\end{itemize}
\textbf{Joint probability density function}
A joint probability density function is a probability distribution
defined for more than one random variable at a time. For two random
variables, \(X\) and \(Z\), we could write their joint density function
as \(f_{X,Z}(x, z)\) , or \(f(x,z)\) for short. The joint density
function encodes all sorts of fun information, including \emph{marginal}
distributions for \(X\) and \(Z\), and conditional distributions (see
next \textbf{bold} definition). We can think of the joint pdf as listing
all possible pairs of outputs from the density function \(f(x,z)\), for
varying values of \(x\) and \(z\). Key things I want you to know about
joint pdfs:
\begin{itemize}
\item
How to get a marginal pdf from a joint pdf:
Suppose I want to know \(f_X(x)\), and I know \(f_{X,Z}(x,z)\). Then I
can integrate or ``average over'' \(Z\) to get
\[
f_X(x) = \int f_{X,Z}(x,z)dz
\]
\item
The relationship between conditional pdfs, marginal pdfs, joint pdfs,
and Bayes' theorem/rule
\item
How to obtain a joint pdf for \emph{independent} random variables:
just multiply their marginal pdfs together! This is how we will
(typically) think about likelihoods!
\item
How to obtain a marginal pdf from a joint pdf when random variables
are independent \emph{without integrating} (think, ``separability'')
\end{itemize}
\textbf{Conditional probability density function}
A conditional pdf denotes the probability distribution for a (set of)
random variable(s), \emph{given that} the value for another (set of)
random variable(s) is known. For two random variables, \(X\) and \(Z\),
we could write the conditional distribution of \(X\) ``given'' \(Z\) as
\(f_{X \mid Z}(x \mid z)\) , where the ``conditioning'' is denoted by a
vertical bar (in LaTeX, this is typeset using ``\textbackslash mid'').
Key things I want you to know about conditional pdfs:
\begin{itemize}
\item
The relationship between conditional pdfs, marginal pdfs, joint pdfs,
and Bayes' theorem/rule
\item
How to obtain a conditional pdf from a joint pdf (again, think Bayes'
rule)
\item
Relationship between conditional pdfs and independence (see next
\textbf{bold} definition)
\end{itemize}
\textbf{Independence}
Two random variables \(X\) and \(Z\) are \emph{independent} if and only
if:
\begin{itemize}
\item
\(f_{X,Z}(x,z) = f_X(x) f_Z(z)\) (their joint pdf is ``separable'')
\item
\(f_{X\mid Z}(x\mid z) = f_X(x)\) (the pdf for \(X\) does not depend
on \(Z\) in any way)
Note that the ``opposite'' is also true:
\(f_{Z\mid X}(z\mid x) = f_Z(z)\)
\end{itemize}
In notation, we denote that two variables are independent as
\(X \perp\!\!\!\perp Z\), or \(X \perp Z\). In LaTeX, the \emph{latter}
is typeset as ``\textbackslash perp'', and the former is typeset as
``\textbackslash perp\textbackslash!\textbackslash!\textbackslash!\textbackslash perp''.
As a matter of personal preference, I (Taylor) prefer
\(\perp\!\!\!\perp\), but I don't like typing it out every time.
Consider using the ``\textbackslash newcommand'' functionality in LaTeX
to create a shorthand for this for your documents!
\textbf{Jacobian Matrix}
Let \(f\) be a 1-1 and onto function, where \(f(x_i) = y_i\) for
\(i = 1, \dots, n\). Then the Jacobian matrix of \(f\) is the matrix of
partial derivatives,
\[
J_f(x) = \begin{pmatrix}
\frac{\partial y_1}{\partial x_1} & \dots & \frac{\partial y_n}{\partial x_1} \\
\vdots & & \vdots \\
\frac{\partial y_1}{\partial x_n} & \dots & \frac{\partial y_n}{\partial x_n}
\end{pmatrix}
\]
The Jacobian matrix is sometimes simply referred to as the ``Jacobian'',
but be careful when simply calling it the Jacobian, since this can
sometimes refer to the \emph{determinant} of the Jacobian matrix as
well. For this course, we'll always refer to the Jacobian matrix as a
matrix, and the ``Jacobian'' as its determinant.
\textbf{Jacobian}
The Jacobian is the determinant of the Jacobian matrix, denoted by
\(| J_f(x) |\). Recall from linear algebra that
\(Det(A) = Det(A^\top)\). This is convenient, because it means we won't
have worry too much about remembering which order our partial
derivatives go in our matrix, for 2x2 matrices (which is all we'll be
working with for this course).
\textbf{Expected Value / Expectation}
The expectation (or expected value) of a random variable is defined as:
\[
E[X] = \int_{-\infty}^\infty x f(x) dx
\]
Expected value is a weighted average, where the average is over all
possible values a random variable can take, weighted by the probability
that those values occur. Key things I want you to know about
expectation:
\begin{itemize}
\item
The relationship between expectation, variance, and moments
(specifically, that \(E[X]\) is the 1st moment!)
\item
The ``law of the unconscious statistician'' (see the Theorems section
of this chapter)
\item
Expectation of linear transformations of random variables (see
\textbf{Theorems} section of this chapter)
\end{itemize}
\textbf{Variance}
The variance of a random variable is defined as:
\[
Var[X] = E[(X - E[X])^2] = E[X^2] - E[X]^2
\]
In words, we can read this as ``the expected value of the squared
deviation from the mean'' of a random variable \(X\). Key things I want
you to know about variance:
\begin{itemize}
\item
The relationship between expectation, variance, and moments (hopefully
clear, given the formula for variance)
\item
The relationship between variance and standard deviation:
\(Var(X) = sd(X)^2\)
\item
The relationship between variance and covariance:
\(Var(X) = Cov(X, X)\)
\item
\(Var(X) \geq 0\). This should make sense, given that we're taking the
expectation of something ``squared'' in order to calculate it!
\item
\(Var(c) = 0\) for any constant, \(c\).
\item
Variance of linear transformations of random variables (see
\textbf{Theorems} section of this chapter)
\end{itemize}
\(r^{th}\) \textbf{moment}
The \(r^{th}\) moment of a probability distribution is given by
\(E[X^r]\). For example, when \(r = 1\), the \(r^{th}\) moment is just
the expectation of the random variable \(X\). Key things I want you to
know about moments:
\begin{itemize}
\item
The relationship between moments, expectation, and variance
\begin{itemize}
\tightlist
\item
For example, if you know the first and second moments of a
distribution, you should be able to calculate the variance of a
random variable with that distribution!
\end{itemize}
\item
The relationship between moments and \emph{moment generating
functions} (see \textbf{Theorems} section of this chapter)
\end{itemize}
\textbf{Covariance}
The covariance of two random variables is a measure of their
\emph{joint} variability. We denote the covariance of two random
variables \(X\) and \(Z\) as \(Cov(X,Z)\), and
\[
Cov(X, Z) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]
\]
Some things I want you to know about covariance:
\begin{itemize}
\item
\(Cov(X, X) = Var(X)\)
\item
\(Cov(X, Y) = Cov(Y, X)\) (order doesn't matter)
\end{itemize}
\textbf{Moment Generating Function (MGF)}
The moment generating function of a random variable \(X\) is defined as
\[
M_X(t) = E[e^{tX}]
\]
A few things to note:
\begin{itemize}
\item
\(M_X(0) = 1\), always.
\item
If two random variables have the same MGF, they have the same
probability distribution!
\item
MGFs are sometimes useful for showing how different random variables
are related to each other
\end{itemize}
\subsection{Distributions Table}\label{distributions-table}
You are also expected to know the probability distributions contained in
Table 1, below. Note that you \emph{do not} need to memorize the pdfs
for these distributions, but you \emph{should} be familiar with what
types of random variables (continuous/quantitative, categorical,
integer-valued, etc.) may take on different distributions. The more
familiar you are with the forms of the pdfs, the easier/faster it will
be to work through problem sets and quizzes.
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}}
>{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}}
>{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}}
>{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2500}}@{}}
\caption{\emph{Table 1.} Table of main probability distributions we will
work with for \emph{MATH/STAT 355}.}\tabularnewline
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Distribution
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
PDF/PMF
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Parameters
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Support
\end{minipage} \\
\midrule\noalign{}
\endfirsthead
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Distribution
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
PDF/PMF
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Parameters
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Support
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Uniform & \(\pi(x) = \frac{1}{\beta - \alpha}\) &
\(\alpha \in \mathbb{R}\), \(\beta\in \mathbb{R}\) &
\(x \in [\alpha, \beta]\) \\
Normal &
\(\pi(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp(-\frac{1}{2\sigma^2} (x - \mu)^2)\)
& \(\mu \in \mathbb{R}\), \(\sigma > 0\) & \(x \in \mathbb{R}\) \\
Multivariate Normal &
\(\pi(\textbf{x}) = (2\pi)^{-k/2} |\Sigma|^{-1/2} \exp(-\frac{1}{2}(\textbf{x} - \mu)^\top \Sigma^{-1}(\textbf{x} - \mu)))\)
& \(\mu \in \mathbb{R}^k\), \(\Sigma \in \mathbb{R}^{k\times k}\) ,
positive semi-definite (in practice, almost always positive definite) &
\(x \in \mathbb{R}^{k}\) \\
Gamma &
\(\pi(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha - 1} e^{-\beta x}\)
& \(\alpha \text{ (shape)}, \beta \text{ (rate)} > 0\) &
\(x \in (0,\infty)\) \\
Chi-squared &
\(\pi(x) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{\nu/2 - 1}e^{-x/2}\) &
\(\nu > 0\) & \(x \in [0, \infty)\) \\
\(F\) &
\(\pi(x) = \frac{\Gamma(\frac{\nu_1 + \nu_2}{2})}{\Gamma(\frac{\nu_1}{2}) \Gamma(\frac{\nu_2}{2})} \left( \frac{\nu_1}{\nu_2}\right)^{\nu_1/2} \left( \frac{x^{(\nu_1 - 2)/2}}{\left( 1 + \left( \frac{\nu_1}{\nu_2}\right)x\right)^{(\nu_1 + \nu_2)/2}}\right)\)
& \(\nu_1, \nu_2 > 0\) & \(x \in [0, \infty)\) \\
Exponential & \(\pi(x) = \beta e^{-\beta x}\) & \(\beta > 0\) &
\(x \in [0, \infty)\) \\
Laplace (Double Exponential) &
\(\pi(x) = \frac{1}{2b} \exp( - \frac{|x - \mu|}{b})\) &
\(\mu \in \mathbb{R}\), \(b > 0\) & \(x \in \mathbb{R}\) \\
Student-\(t\) &
\(\pi(x) = \frac{\Gamma((\nu + 1)/2)}{\Gamma(\nu/2) \sqrt{\nu \pi}} (1 + \frac{x^2}{\nu})^{-(\nu + 1)/2}\)
& \(\nu > 0\) & \(x \in \mathbb{R}\) \\
Beta &
\(\pi(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1}(1 - x)^{\beta - 1}\)
& \(\alpha, \beta > 0\) & \(x \in [0,1]\) \\
Poisson & \(\pi(x) = \frac{\lambda^x e^{-\lambda}}{x!}\) &
\(\lambda > 0\) & \(x \in \mathbb{N}\) \\
Binomial & \(\pi(x) = \binom{n}{x} p^{x} (1 - p)^{n - x}\) &
\(p \in [0,1], n = \{0, 1, 2, \dots\}\) &
\(x \in \{0, 1, \dots, n\}\) \\
Multinomial &
\(\pi(\textbf{x}) = \frac{n!}{x_1! \dots x_k!} p_1^{x_1} \dots p_k^{x_k}\)
& \(p_i > 0\), \(p_1 + \dots + p_k = 1\), \(n = \{0, 1, 2, \dots \}\) &
\(\{ x_1, \dots, x_k \mid \sum_{i = 1}^k x_i = n, x_i \geq 0 (i = 1, \dots, k)\}\) \\
Negative Binomial & \(\pi(x) = \binom{x + r - 1}{x} (1-p)^x p^r\) &
\(r > 0\), \(p \in [0,1]\) & \(x \in \{0, 1, \dots\}\) \\
\end{longtable}
\section{Theorems}\label{theorems}
\begin{itemize}
\item
Law of Total Probability
\[
P(A) = \sum_n P(A \cap B_n),
\]or
\[
P(A) = \sum_n P(A \mid B_n) P(B_n)
\]
\item
Bayes' Theorem
\[
\pi(A \mid B) = \frac{\pi(B \mid A) \pi(A)}{\pi(B)}
\]
\item
Relationship between pdf and cdf
\[
F_Y(y) = \int_{-\infty}^y f_Y(t)dt
\]
\[
\frac{\partial}{\partial y}F_Y(y) = f_Y(y)
\]
\item
Expectation of random variables
\[
E[X] = \int_{-\infty}^\infty x f(x) dx
\]
\[
E[X^2] = \int_{-\infty}^\infty x^2 f(x) dx
\]
\begin{itemize}
\item
``Law of the Unconscious Statistician''
\[
E[g(X)] = \int_{-\infty}^\infty g(x)f(x)dx
\]
\end{itemize}
\item
Expectation and variance of linear transformations of random variables
\[
E[cX + b] = c E[X] + b
\]
\[
Var[cX + b] = c^2 Var[X]
\]
\item
Relationship between mean and variance
\[
Var[X] = E[(X - E[X])^2] = E[X^2] - E[X]^2
\]
Also, recall that \(Cov[X, X] = Var[X]\).
\item
Iterated Things
\end{itemize}
\[
E[X] = E[E[X \mid Y]]
\] and \[
Var(X) = E[Var(X \mid Y)] + Var(E[X \mid Y])
\]
\begin{itemize}
\item
Finding a marginal pdf from a joint pdf
\[
f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x, y) dy
\]
\item
Independence of random variables and joint pdfs
If two random variables are independent, their joint pdf will be
\emph{separable}. For example, if \(X\) and \(Y\) are independent, we
could write
\[
f_{X,Y}(x, y) = f_{X}(x)f_Y(y)
\]
\item
Expected value of a product of independent random variables
Suppose random variables \(X_1, \dots, X_n\) are independent. Then we
can write,
\[
E\left[\prod_{i = 1}^n X_i\right] = \prod_{i = 1}^n E[X_i]
\]
\item
Covariance of independent random variables
If \(X\) and \(Y\) are independent, then \(Cov(X, Y) = 0\). We can
show this by noting that
\end{itemize}
\begin{align}
Cov(X, Y) & = E[(X - E[X])(Y - E[Y])] \\
& = E[XY - XE[Y] - YE[X] + E[X]E[Y]] \\
& = E[XY] - E[XE[Y]] - E[YE[X]] + E[X]E[Y] \\
& = 2E[X]E[Y] - 2E[X]E[Y] \\
& = 0
\end{align}
\begin{itemize}
\item
Using MGFs to find moments
Recall that the moment generating function of a random variable \(X\),
denoted by \(M_X(t)\) is
\[
M_X(t) = E[e^{tX}]
\]
Then the \(n\)th moment of the probability distribution for \(X\) ,
\(E[X^n]\), is given by
\[
\frac{\partial M_X}{\partial t^n} \Bigg|_{t = 0}
\]
where the above reads as ``the \(n\)th derivative of the moment
generating function, evaluated at \(t = 0\).''
\item
Using MGFs to identify pdfs
MGFs uniquely identify probability density functions. If \(X\) and
\(Y\) are two random variables where for all values of \(t\),
\(M_X(t) = M_Y(t)\), then \(F_X(x) = F_Y(y)\).
\item
Central Limit Theorem
The classical CLT states that for independent and identically
distributed (iid) random variables \(X_1, \dots, X_n\), with expected
value \(E[X_i] = \mu\) and \(Var[X_i] = \sigma^2 < \infty\), the
sample average (centered and standardized) converges in distribution
to a standard normal distribution at a root-\(n\) rate. Notationally,
this is written as
\[
\sqrt{n} (\bar{X} - \mu) \overset{d}{\to} N(0, \sigma^2)
\]
\includegraphics[width=0.20833in,height=0.16667in]{images/chilipepper.png}
A fun aside: this is only \emph{one} CLT, often referred to as the
Levy CLT. There are other CLTs, such as the Lyapunov CLT and
Lindeberg-Feller CLT!
\end{itemize}
\subsection{Transforming Continuous Random
Variables}\label{transforming-continuous-random-variables}
We will \emph{often} take at face value previously proven
\emph{relationships} between random variables. What I mean by this, as
an example, is that it is a nice (convenient) fact that a sum of two
independent normal random variables is \emph{still} normally
distributed, with a nice form for the mean and variance. In particular,
if \(X \sim N(\mu, \sigma^2)\) and \(Y \sim N(\theta, \nu^2)\), then
\(X + Y \sim N(\mu + \theta, \sigma^2 + \nu^2)\). Most frequently used
examples of these sorts of relationships can be found in the ``Related
Distributions'' section of the Wikipedia page for a given probability
distribution. Unless I explicitly ask you to derive/show how certain
variables are related to each other, you can just state the known
relationship, use it, and move on!
If I \emph{do} ask you to derive/show these things, there a few
different ways we can go about this. For this course, I expect you to
know the ``CDF method'' for \emph{one function} of \emph{one random
variable}, \textbf{and} the ``Jacobian method'' for a function of
\emph{more than one} random variable.
\subsubsection{CDF Method: One random
variable}\label{cdf-method-one-random-variable}
\textbf{Theorem}. Let \(X\) be a continuous random variable with pdf
\(f_X(x)\). Define a new random variable \(Y = g(X)\), for nice*
functions \(g\). Then
\(f_Y(y) = f_X(g^{-1}(y)) \times \frac{1}{g'(g^{-1}(y))}\).
\includegraphics[width=0.20833in,height=0.16667in]{images/chilipepper.png}
*By \emph{nice} functions we mean functions that are strictly increasing
and smooth \emph{on the required range}. As an example, \(exp(x)\) is a
smooth, strictly increasing function; \(|x|\) is not on the \emph{whole
real line}, but \emph{is} from \((0, \infty)\) (where a lot of useful
pdfs are defined). For the purposes of this class, every function that
you will need to do this for will be ``nice.'' Note that there are also
considerations that need to be taken regarding the \emph{range} of
continuous random variables when considering transforming them. We will
mostly ignore these considerations in this class, but a technically
complete derivation (or proof) must consider them.
Proof.
We can write
\begin{align*}
f_Y(y) & = \frac{\partial}{\partial y} F_Y(y) \\
& = \frac{\partial}{\partial y} \Pr(Y \leq y) \\
& = \frac{\partial}{\partial y} \Pr(g(X) \leq y) \\
& = \frac{\partial}{\partial y} \Pr(X \leq g^{-1}(y)) \\
& = \frac{\partial}{\partial y} F_X(g^{-1}(y)) \\
& = f_X(g^{-1}(y)) \times \frac{\partial}{\partial y} g^{-1}(y)
\end{align*}
where to obtain the last equality we use chain rule! Now we require some
statistical trickery to continue\ldots{} (note that this method is
called the ``CDF method'' because we go \emph{through} the CDF to derive
the distribution for \(Y\))
You will \emph{especially} see this in the Bayes chapter of our course
notes, but it is often true that our lives are made easier as
statisticians if we multiply things by one, or add zero. What exactly do
I mean? Rearranging gross looking formulas into things we are familiar
with (like pdfs, for example) often makes our lives easier and allows us
to avoid dealing with such grossness. Here, the grossness is less
obvious, but nonetheless relevant. Note that we can write
\begin{align*}
y & = y \\
y & = g(g^{-1}(y)) \\
\frac{\partial}{\partial y} y & = \frac{\partial}{\partial y} g(g^{-1}(y)) \\
1 & = g'(g^{-1}(y)) \frac{\partial}{\partial y} g^{-1}(y) \hspace{1cm} \text{(chain rule again!)} \\
\frac{1}{g'(g^{-1}(y))} & = \frac{\partial}{\partial y} g^{-1}(y)
\end{align*}
The right-hand side should look familiar: it is exactly what we needed
to ``deal with'' in our proof! Returning to that proof, we have
\begin{align*}
f_Y(y) & = f_X(g^{-1}(y)) \times \frac{\partial}{\partial y} g^{-1}(y) \\
& = f_X(g^{-1}(y)) \times \frac{1}{g'(g^{-1}(y))}
\end{align*}
as desired.
\subsubsection{Jacobian Method: More than one random
variable}\label{jacobian-method-more-than-one-random-variable}
Transformations of single random variables are great, but we'll need to
work with transformations of \emph{more than one} random variable if we
want to be able to manipulate joint pdfs. Suppose, for example, we have
\(X \sim Gamma(\alpha, \lambda)\) and \(Y \sim Gamma(\beta, \lambda)\),
where \(X \perp\!\!\!\perp Y\). Let \(U = X + Y\) and
\(W = \frac{X}{X + Y}\). How do we show that \(U\) and \(W\) are
independent? Through finding the joint pdf! Which means we need a method
for transforming more than one, continuous random variable. Enter the
Jacobian Method.
\textbf{Theorem}. Let \(X = (x_1, \dots, x_n)\) be a vector of random
variables (a random vector) with pdf \(\pi_{X}(x)\). Suppose that
\(f(x) = y\) is a smooth, 1-1 and onto function, and that
\(|J_f(x)| > 0\) almost everywhere. Then \(\pi_Y(y)\) is given by
\[
\pi_Y(y) = \pi_{X}(f^{-1}(y)) \times \bigg| \frac{\partial x}{\partial y} \bigg| \times I_Y\{y\}
\] where \(I_Y \{y \}\) denotes the support of \(Y\).
NOTE: \(| J_f(y) | = | \frac{\partial x}{\partial y} |\), above, where
\(f(x) = y\). This means that the numerators and denominators in the
Jacobian Matrix in the definition in the Course Notes are \emph{flipped}
here. The reason for this becomes clear in the proof sketched below,
noting that
\(| \frac{\partial x}{\partial y} |^{-1} = | \frac{\partial y}{\partial x} |\),
for the supposed functions \(f\).
Proof.
Note that if \(f\) is both 1-1 and onto, then \(f\) is \emph{either}
monotone increasing or monotone decreasing. We'll prove the case where
\(f\) is increasing, and we'll note (but not show) how the decreasing
case follows directly.
Let \(f\) be a smooth, monotone increasing function. For some subset
\(C \subseteq \mathbb{R}^n\),
\begin{align*}
\int_C \pi_Y(y) dy & = \Pr(Y \in C) \\
& = \Pr(f(X) \in C) \\
& = \Pr(X \in f^{-1}(C)) \\
& = \int_{f^{-1}(C)} \pi_X(x) dx
\end{align*}
Now we'll use a (convoluted) \(u\)-substitution to make this look like