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\title{Transformations of Jointly Distributed Random Variables\footnote{ This slide show is an open-source document. See last slide for copyright information.}}
\subtitle{STA 256: Fall 2019}
\date{} % To suppress date
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}
\begin{frame}
\frametitle{Transformations of Jointly Distributed Random Variables} \pause
%\framesubtitle{}
Let $Y = g(X_1, \ldots, X_n)$. What is the probability distribution of~$Y$? \pause
For example,
\begin{itemize}
\item $X_1$ is the number of jobs completed by employee 1.
\item $X_2$ is the number of jobs completed by employee 2.
\item You know the probability distributions of $X_1$ and $X_2$. \pause
\item You would like to know the probability distribution of $Y = X_1 + X_2$.
\end{itemize}
\end{frame}
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\section{Convolutions}
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\begin{frame}
\frametitle{Convolutions of discrete random variables} \pause
%\framesubtitle{}
\begin{itemize}
\item Let $X$ and $Y$ be discrete random variables.
\item The standard case is where they are independent.
\item Want probability mass function of $Z = X + Y$.
\end{itemize} \pause
\begin{eqnarray*}
p_{_Z}(z) & = & P(Z=z) \\ \pause
& = & P(X+Y=z) \\ \pause
& = & \sum_x P(X+Y=z|X=x)P(X=x) \\ \pause
& = & \sum_x P(x+Y=z|X=x)P(X=x) \\ \pause
& = & \sum_x P(Y=z-x|X=x)P(X=x) \\ \pause
& = & \sum_x P(Y=z-x)P(X=x) \mbox{ by independence}\\ \pause
& = & \sum_x p_{_X}(x) p_{_Y}(z-x)
\end{eqnarray*}
\end{frame}
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\begin{frame}
\frametitle{Summarizing}
\framesubtitle{Convolutions of discrete random variables}
Let $X$ and $Y$ be \emph{independent} discrete random variables, and $Z = X + Y$.
{\LARGE
\begin{displaymath}
p_{_Z}(z) = \sum_x p_{_X}(x) p_{_Y}(z-x)
\end{displaymath}
} % End size
\end{frame}
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\begin{frame}
\frametitle{Two Important results}
\framesubtitle{Proved using the convolution formula} \pause
\begin{itemize}
\item Let $X \sim$ Poisson($\lambda_1$) and $Y \sim$ Poisson($\lambda_2$) be independent. \pause Then $Z=X+Y \sim$ Poisson($\lambda_1+\lambda_2$). \pause % Sum using binomial theorem
\item Let $X \sim$ Binomial($n_1,\theta$) and $Y \sim$ Binomial($n_2,\theta$) be independent. \pause Then $Z=X+Y \sim$ Binomial($n_1+n_2,\theta$) % Sum over a hypergeometric.
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Convolutions of \emph{continuous} random variables} \pause
%\framesubtitle{}
\begin{itemize}
\item Let $X$ and $Y$ be continuous random variables.
\item The standard case is where they are independent.
\item Want probability density function of $Z = X + Y$.
\end{itemize} \pause
\begin{columns}
\column{0.5\textwidth}
\begin{eqnarray*}
f_{_Z}(z) & = & \frac{d}{dz} P(Z \leq z) \\ \pause
& = & \frac{d}{dz} P(X+Y \leq z) \\
&&\\ &&\\ &&\\ &&\\ &&\\
\pause
\end{eqnarray*}
\column{0.5\textwidth}
%\begin{center}
\includegraphics[width=2in]{x+y0$. \pause Then \\ $Z=X+Y \sim$ Gamma($\alpha=2,\lambda$). \pause % Integrate cdf of exponential.
\item Let $X \sim$ Normal($\mu_1,\sigma^2_1$) and $Y \sim$ Normal($\mu_2,\sigma^2_2$) be independent. \pause Then \\
$Z=X+Y \sim$ Normal$\left(\mu_1+\mu_2,\sigma^2_1+\sigma^2_2\right)$. % Complete the square, ugh!
\end{itemize}
\end{frame}
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\section{Jacobians}
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\begin{frame}
\frametitle{The Jacobian Method} \pause
%\framesubtitle{}
\begin{itemize}
\item $X_1$ and $X_2$ are continuous random variables. \pause
\item $Y_1 = g_1(X_1,X_2)$ and $Y_2 = g_2(X_1,X_2)$. \pause
\item Want $f_{_{Y_1,Y_2}}(y_1,y_2)$ \pause
\end{itemize}
Solve for $x_1$ and $x_2$, obtaining $x_1(y_1,y_2)$ and $x_2(y_1,y_2)$\pause. Then
\begin{displaymath}
f_{_{Y_1,Y_2}}(y_1,y_2) = f_{_{X_1,X_2}}(\, x_1(y_1,y_2),x_2(y_1,y_2) \,) \pause \cdot abs
\renewcommand{\arraystretch}{1.5}
\left| \begin{array}{cc}
\frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2} \\
\frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2}
\end{array}\right|
\renewcommand{\arraystretch}{1.0}
\end{displaymath} \pause
The determinant
$\left| \begin{array}{cc}
a & b \\
c & d
\end{array}\right| = ad-bc$.
\end{frame}
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\begin{frame}
\frametitle{More about the Jacobian method}
\framesubtitle{$Y_1 = g_1(X_1,X_2)$ and $Y_2 = g_2(X_1,X_2)$} \pause
\begin{itemize}
\item It follows directly from a change of variables formula in multi-variable integration. The proof is omitted. \pause
\item It must be possible to solve $y_1 = g_1(x_1,x_2)$ and $y_2 = g_2(x_1,x_2)$ for $x_1$ and $x_2$. \pause
\item That is, the function $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ must be one to one (injective). \pause
\item Frequently you are only interested in $Y_1$, and $Y_2 = g_2(X_1,X_2)$ is chosen to make reverse solution easy. \pause
\item The partial derivatives must all be continuous, except possibly on a set of probability zero (they almost always are). \pause
\item It extends naturally to higher dimension.
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Change from rectangular to polar co-ordinates}
\framesubtitle{By the Jacobian method} \pause
A point on the plane may be represented as $(x,y)$, or \pause
\begin{center}
\includegraphics[width=2.5in]{circle}
\end{center}
An angle $\theta$ and a radius $r$.
\end{frame}
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\begin{frame}
\frametitle{Change of variables}
\framesubtitle{From rectangular to polar coordinates}
\begin{columns}
\column{0.6\textwidth}
\begin{center}
\includegraphics[width=2.5in]{circle}
\end{center} \pause
\column{0.4\textwidth}
\begin{itemize}
\item[] $x = r \cos(\theta)$
\item[] $y = r \sin(\theta)$ \pause
\item[] $x^2 + y^2 = r^2$ \pause
\end{itemize}
\begin{itemize}
\item As $x$ and $y$ range from $-\infty$ to $\infty$, \pause
\item $r$ goes from 0 to $\infty$
\item And $\theta$ goes from $0$ to $2\pi$.
\end{itemize}
\end{columns}
\end{frame}
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\begin{frame}
\frametitle{Integral $\int_0^\infty \int_0^\infty f_{x,y}(x,y) \, dx \, dy$} \pause
%\framesubtitle{}
Change of variables: \pause
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item[] $x = r \cos(\theta)$
\item[] $y = r \sin(\theta)$
\end{itemize}
\column{0.5\textwidth}
\begin{center}
\includegraphics[width=1.75in]{circle}
\end{center}
\end{columns}\pause
\begin{eqnarray*}
& & \int_0^\infty \int_0^\infty f_{x,y}(x,y) \, dx \, dy \\
&=& \pause \int_0^{\pi/2} \int_0^\infty f_{x,y}(r\cos\theta,r\sin\theta) \, abs
\renewcommand{\arraystretch}{1.5}
\left| \begin{array}{cc}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}
\end{array}\right|
\renewcommand{\arraystretch}{1.0}
\, dr \, d\theta
\end{eqnarray*}
\end{frame}
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\begin{frame}
\frametitle{Evaluate the determinant}
\framesubtitle{(with $x = r\cos(\theta)$ and $y = r\sin(\theta)$)}
\begin{eqnarray*}
\renewcommand{\arraystretch}{1.5}
\left| \begin{array}{cc}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}
\end{array}\right|
\renewcommand{\arraystretch}{1.0} \pause
& = &
\renewcommand{\arraystretch}{1.5}
\left| \begin{array}{cc}
\frac{\partial \, r\cos(\theta)}{\partial r} & \frac{\partial \, r\cos(\theta)}{\partial \theta} \\
\frac{\partial \, r\sin(\theta)}{\partial r} & \frac{\partial \, r\sin(\theta)}{\partial \theta}
\end{array}\right|
\renewcommand{\arraystretch}{1.0} \pause
\\
&& \\
& = &
\left| \begin{array}{cc}
\cos(\theta) & -r\sin(\theta) \\
\sin(\theta) & r\cos(\theta)
\end{array}\right| \pause
\\
&& \\
& = & r \cos^2\theta - - r\sin^2\theta \pause \\
& = & r (\sin^2\theta + \cos^2\theta) \pause \\
& = & r
\end{eqnarray*}
\end{frame}
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\begin{frame}
\frametitle{So the integral is} \pause
%\framesubtitle{}
{\large
\begin{displaymath}
\int_0^\infty \int_0^\infty f_{x,y}(x,y) \, dx \, dy =
\int_0^{\pi/2} \int_0^\infty f_{x,y}(r\cos\theta,r\sin\theta) \, r \, dr \, d\theta
\end{displaymath} \pause
} % End size
\begin{itemize}
\item The standard formula for change from rectangular to polar co-ordinates is $dx \, dy = r \, dr \, d\theta$. \pause
\item It comes from a Jacobian. \pause
\item Other limits of integration are possible. \pause
\item $f(x,y)$ does not have to be a density.
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Copyright Information}
This slide show was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
{Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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\end{document}
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\frametitle{} \pause
%\framesubtitle{}
\begin{itemize}
\item \pause
\item \pause
\item
\end{itemize}
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\begin{center}
\includegraphics[width=2in]{BivariateNormal}
\end{center}
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