\documentclass[notitlepage]{article} \usepackage{../Math323} \usepackage{listings} \usepackage{numprint} \usepackage{enumerate} \usepackage{multirow} \usepackage{bbm} \usepackage{amsfonts,amsmath} \setlength{\parindent}{0pt} \def\E{\Expect} \lstloadlanguages{R} \definecolor{keywordcolor}{rgb}{0,0.6,0.6} \definecolor{delimcolor}{rgb}{0.461,0.039,0.102} \definecolor{Rcommentcolor}{rgb}{0.101,0.043,0.432} \lstdefinestyle{Rsettings}{ basicstyle=\ttfamily, breaklines=true, showstringspaces=false, keywords={if, else, function, theFunction, tmp}, % Write as many keywords otherkeywords={}, commentstyle=\itshape\color{Rcommentcolor}, keywordstyle=\color{keywordcolor}, moredelim=[s][\color{delimcolor}]{"}{"}, } \lstset{basicstyle=\ttfamily, numbers=none, literate={~} {$\sim$}{2}} \begin{document} <>= library(knitr) # global chunk options opts_chunk$set(cache=TRUE, autodep=TRUE) options(scipen=999) options(repos=c(CRAN="https://cloud.r-project.org/")) inline_hook <- function (x) { if (is.numeric(x)) { # ifelse does a vectorized comparison # If integer, print without decimal; otherwise print two places res <- ifelse(x == round(x), sprintf("%.3f", x), sprintf("%.3f", x) ) paste(res, collapse = ", ") } } knit_hooks$set(inline = inline_hook) @ \begin{center} {\textsclarge{Joint Continuous Distributions}} \end{center} \textbf{Joint cdf:} If $Y_1$ and $Y_2$ are continuous random variables, then we may consider the probability \[ F_{Y_1,Y_2}(y_1,y_2) = P((Y_1 \leq y_1) \cap (Y_2 \leq y_2)) = P (Y_1 \leq y_1, Y_2 \leq y_2) \] for any pair of real numbers $(y_1,y_2)$. The joint cdf has the following properties: \begin{itemize} \item ``starts at zero'' \begin{align*} \lim_{y_1 \lra -\infty} \lim_{y_2 \lra -\infty} F_{Y_1,Y_2}(y_1,y_2) & = 0 \\[6pt] \lim_{y_1 \lra -\infty} F_{Y_1,Y_2}(y_1,y_2) & = 0 \quad \textrm{for all } y_2 \qquad \text{and} \qquad \lim_{y_2 \lra -\infty} F_{Y_1,Y_2}(y_1,y_2) = 0 \quad \textrm{for all } y_1 \end{align*} \item ``ends at one'' \[ \lim_{y_1 \lra \infty} \lim_{y_2 \lra \infty} F_{Y_1,Y_2}(y_1,y_2) = 1 \] \item ``non-decreasing in $y_1$ and $y_2$ in between'' \begin{align*} F_{Y_1,Y_2}(y_1,y_2) & \leq F_{Y_1,Y_2}(y_1+c,y_2) \\[6pt] F_{Y_1,Y_2}(y_1,y_2) & \leq F_{Y_1,Y_2}(y_1,y_2+c) \end{align*} for all $y_1,y_2$, and any $c > 0$. \end{itemize} Furthermore, we have that \[ \lim_{y_1 \lra \infty} F_{Y_1,Y_2}(y_1,y_2) = F_{Y_2}(y_2) \qquad \qquad \lim_{y_2 \lra \infty} F_{Y_1,Y_2}(y_1,y_2) = F_{Y_1}(y_1). \] \bigskip \textbf{EXAMPLE:} Consider the function defined \begin{itemize} \item for $0 \leq y_1 \leq 1, 0 \leq y_2 \leq 1$ by \[ F_{Y_1,Y_2}(y_1,y_2) = \frac{4}{5} \left (y_1 y_2 + \frac{y_1^2 y_2^2}{4} \right) \] \item for $y_1 < 0$ or $y_2 < 0$ by \[ F_{Y_1,Y_2}(y_1,y_2) = 0 \] \item for $y_1 > 1$, but $0 \leq y_2 \leq 1$ by \[ F_{Y_1,Y_2}(y_1,y_2) = F_{Y_1,Y_2}(1,y_2) = \frac{4}{5} \left (y_2 + \frac{y_2^2}{4} \right) \] which follows as \[ P(Y_1 \leq y_1, Y_2 \leq y_2) = P(Y_1 \leq 1, Y_2 \leq y_2). \] \item for $y_2 > 1$, but $0 \leq y_1 \leq 1$ by \[ F_{Y_1,Y_2}(y_1,y_2) = F_{Y_1,Y_2}(y_1,1) = \frac{4}{5} \left (y_1 + \frac{y_1^2}{4} \right) \] which follows as in this region \[ P(Y_1 \leq y_1, Y_2 \leq y_2) = P(Y_1 \leq y_1, Y_2 \leq 1). \] \item for $y_1 > 1$ and $y_2 > 1$ by $F_{Y_1,Y_2}(y_1,y_2) = 1$. \end{itemize} This function defines a joint cdf for two continuous random variables $Y_1$ and $Y_2$ taking values on $[0,1] \times [0,1]$. \vspace*{-0.5 in} <>= par(pty='s') library(fields,quietly=TRUE) Ffunc<-function(x,y){ xv<-pmin(1,x) yv<-pmin(1,y) return(0.8*(xv*yv + (xv*yv/2)^2)*(xv >= 0 & yv >= 0)) } y1<-y2<-sort(c(seq(-0.25,1.25,by=0.05),-0.01,0.01)) F12<-outer(y1,y2,Ffunc) z<-F12 colfunc <- colorRampPalette(c("blue","lightblue","white","yellow","orange","red")) z.facet.center <- (z[-1, -1] + z[-1, -ncol(z)] + z[-nrow(z), -1] + z[-nrow(z), -ncol(z)])/4 z.facet.range<-cut(z.facet.center, 100) colvec<-colfunc(100)[z.facet.range] par(mar=c(0,2,0,0)) persp(y1,y2,F12,col=colvec,phi=20,theta=-60, ticktype="detailed",expand=0.6,xlab="y1",ylab="y2",zlab="F(y1,y2)",cex.axis=0.6) @ \vspace*{-0.5 in} This surface plot displays the function $F_{Y_1,Y_2}(y_1,y_2)$ on a grid of values. The function is zero in the dark blue region, and increases in both $y_1$ and $y_2$ for values in the range $(0,1)$. It is also possible to plot the function as an image plot (see below), with contours linking points on the plot where the function is constant, that is, where $F_{Y_1,Y_2}(y_1,y_2) = p$, for values $p$, $0 \leq p \leq 1$. \vspace*{-0.5 in} <>= par(pty='s') y1<-y2<-seq(-0.25,1.25,by=0.005) F12<-outer(y1,y2,Ffunc) image.plot(y1,y2,F12,col=colfunc(100),xlab=expression(y[1]),ylab=expression(y[2]),axes=F) axis(1, at = seq(0, 1, by = 0.2),cex.axis=0.6) axis(2, at = seq(0, 1, by = 0.2),cex.axis=0.6) contour(y1,y2,F12,add=T) abline(h=1,v=1,lty=2) abline(h=0,v=0,lty=2) @ \bigskip \textbf{Joint pdf:} We then use a similar construction to the single variable case to define the joint pdf, that is, we write \[ F_{Y_1,Y_2}(y_1,y_2) = \int_{-\infty}^{y_1} \int_{-\infty}^{y_2} f_{Y_1,Y_2}(t_1,t_2) \: d t_2 d t_1 \] to define $f_{Y_1,Y_2}(y_1,y_2)$ implicitly. We have then that \[ f_{Y_1,Y_2}(y_1,y_2) = \frac{\partial^2}{\partial y_1 \partial y_2} \left\{F_{Y_1,Y_2}(y_1,y_2)\right\} \] In the above example, we have that for $0 \leq y_1 \leq 1$, $0 \leq y_2 \leq 1$ \begin{align*} f_{Y_1,Y_2}(y_1,y_2) = \frac{\partial^2}{\partial y_1 \partial y_2} \left\{ \frac{4}{5} \left (y_1 y_2 + \frac{y_1^2 y_2^2}{4} \right) \right\} & = \frac{\partial}{\partial y_1} \left\{ \frac{\partial}{\partial y_2} \left\{ \frac{4}{5} \left (y_1 y_2 + \frac{y_1^2 y_2^2}{4} \right) \right\} \right\}\\[6pt] & = \frac{\partial}{\partial y_1} \left\{ \frac{4}{5} \left (y_1 + \frac{y_1^2 y_2 }{2} \right) \right\} \\[6pt] & = \frac{4}{5} \left (1 + y_1 y_2 \right) \\[6pt] \end{align*} Notice that for $y_1$ and $y_2$ outside of the range $0 \leq y_1 \leq 1$, $0 \leq y_2 \leq 1$, we have that \[ f_{Y_1,Y_2}(y_1,y_2) = 0. \] <>= par(pty='s') ffunc<-function(x,y){ xv<-x yv<-y return(0.8*(1+xv*yv)*(xv >= 0 & yv >= 0 & xv <= 1 & yv <= 1)) } y1<-y2<-sort(c(seq(-0.25,1.25,by=0.05),-0.01,0.01)) f12<-outer(y1,y2,ffunc) z<-f12 colfunc <- colorRampPalette(c("blue","lightblue","white","yellow","orange","red")) z.facet.center <- (z[-1, -1] + z[-1, -ncol(z)] + z[-nrow(z), -1] + z[-nrow(z), -ncol(z)])/4 z.facet.range<-cut(z.facet.center, 100) colvec<-colfunc(100)[z.facet.range] par(mar=c(0,2,0,0)) persp(y1,y2,f12,col=colvec,phi=20,theta=-60, ticktype="detailed",expand=0.6,xlab="y1",ylab="y2",zlab="f(y1,y2)",cex.axis=0.6) @ \vspace*{-0.5 in} This surface plot of the joint pdf indicates that the joint pdf in this case is also increasing in $y_1$ and $y_2$ on the range $(0,1)$ for each variable. \vspace*{-0.5 in} <>= par(pty='s') y1<-y2<-seq(-0.25,1.25,by=0.01) f12<-outer(y1,y2,ffunc) image.plot(y1,y2,f12,col=colfunc(100),xlab=expression(y[1]),ylab=expression(y[2]),axes=F) axis(1, at = seq(0, 1, by = 0.2),cex.axis=0.8) axis(2, at = seq(0, 1, by = 0.2),cex.axis=0.8) contour(y1,y2,f12,add=T) abline(h=1,v=1,lty=2) abline(h=0,v=0,lty=2) @ \textbf{Marginal distributions:} we may compute the marginal pdfs by noting that \[ F_{Y_1}(y_1) = F_{Y_1,Y_2}(y_1,1) = \frac{4}{5} \left (y_1 + \frac{y_1^2}{4} \right) \qquad 0 \leq y_1 \leq 1 \] with $F_{Y_1}(y_1) = 0$ if $y_1 < 0$ and $F_{Y_1}(y_1) = 1$ if $y_1 > 0$. Similarly \[ F_{Y_2}(y_2) = F_{Y_1,Y_2}(1,y_2) = \frac{4}{5} \left (y_2 + \frac{y_2^2}{4} \right) \qquad 0 \leq y_2 \leq 1 \] with $F_{Y_2}(y_2) = 0$ if $y_2 < 0$ and $F_{Y_2}(y_2) = 1$ if $y_2 > 0$. By differentiation, we have that \[ f_{Y_1}(y_1) = \left \{ \begin{array}{ll} \dfrac{4}{5} \left (1 + \dfrac{y_1}{2} \right) & 0 \leq y_1 \leq 1\\[12pt] 0 & \text{otherwise} \end{array} \right. . \] Notice that \[ \int_{-\infty}^\infty f_{Y_1}(y_1) \: d y_1 = \int_{0}^1 \frac{4}{5} \left (1 + \frac{y_1}{2} \right) \: d y_1 = \frac{4}{5} \left[y_1 + \frac{y_1^2}{4} \right]_0^1 = \frac{4}{5} \left[1 + \frac{1}{4} \right] = 1. \] By an identical calculation \[ f_{Y_2}(y_2) = \left \{ \begin{array}{ll} \dfrac{4}{5} \left (1 + \dfrac{y_2}{2} \right) & 0 \leq y_2 \leq 1\\[12pt] 0 & \text{otherwise} \end{array} \right. . \] Here we have that for general $(y_1,y_2)$, $f_{Y_1,Y_2}(y_1,y_2) \neq f_{Y_1}(y_1) f_{Y_2}(y_2)$, so therefore the random variables $Y_1$ and $Y_2$ are \textbf{not independent} (that is, they are \textbf{dependent}). \bigskip \textbf{Conditional pdfs:} we can compute the conditional pdfs using the definition. For example, for $0 \leq y_2 \leq 1$ \[ f_{Y_1|Y_2}(y_1|y_2) = \frac{f_{Y_1,Y_2}(y_1,y_2)}{f_{Y_2}(y_2)} = \frac{\left (1 + y_1 y_2 \right)}{\left (1 + \dfrac{y_2}{2} \right)} \qquad 0 \leq y_1 \leq 1, \text{zero otherwise}. \] \bigskip \textbf{EXAMPLE:} Consider the function defined for $y_1 \in \R$ and $y_2 \in \R$ by \[ F_{Y_1,Y_2}(y_1,y_2) = \frac{e^{y_1+y_2}}{(1+e^{y_1})(1+e^{y_2})} \] This function defines a joint cdf for two continuous random variables $Y_1$ and $Y_2$. \vspace*{-0.5 in} <>= par(pty='s') library(fields,quietly=TRUE) Ffunc<-function(x,y){ return((exp(x)/(1+exp(x))*(exp(y)/(1+exp(y))))) } y1<-y2<-seq(-3,3,by=0.1) F12<-outer(y1,y2,Ffunc) z<-F12 colfunc <- colorRampPalette(c("blue","lightblue","white","yellow","orange","red")) z.facet.center <- (z[-1, -1] + z[-1, -ncol(z)] + z[-nrow(z), -1] + z[-nrow(z), -ncol(z)])/4 z.facet.range<-cut(z.facet.center, 100) colvec<-colfunc(100)[z.facet.range] par(mar=c(2,4,1,0)) persp(y1,y2,F12,col=colvec,phi=20,theta=-60, ticktype="detailed",expand=0.6,xlab="y1",ylab="y2",zlab="F(y1,y2)",cex.axis=0.6,zlim=range(0,1)) @ \vspace*{-1.0 in} <>= par(pty='s') y1<-y2<-seq(-3,3,by=0.01) F12<-outer(y1,y2,Ffunc) image.plot(y1,y2,F12,col=colfunc(100),xlab=expression(y[1]),ylab=expression(y[2]), axes=F) axis(1, at = seq(-3, 3, by = 1),cex.axis=0.8) axis(2, at = seq(-3, 3, by = 1),cex.axis=0.8) contour(y1,y2,F12,add=T) @ \textbf{Joint pdf:} We have by direct calculation, differentiating the joint cdf, that \[ f_{Y_1,Y_2}(y_1,y_2) = \frac{e^{y_1+y_2}}{(1+e^{y_1})^2 (1+e^{y_2})^2} = \frac{e^{y_1}}{(1+e^{y_1})^2} \frac{e^{y_2}}{(1+e^{y_2})^2} \] \vspace*{-0.5 in} <>= par(pty='s') ffunc<-function(x,y){ xv<-x yv<-y return((exp(x)/(1+exp(x))^2*(exp(y)/(1+exp(y))^2))) } y1<-y2<-seq(-3,3,by=0.05) f12<-outer(y1,y2,ffunc) z<-f12 colfunc <- colorRampPalette(c("blue","lightblue","white","yellow","orange","red")) z.facet.center <- (z[-1, -1] + z[-1, -ncol(z)] + z[-nrow(z), -1] + z[-nrow(z), -ncol(z)])/4 z.facet.range<-cut(z.facet.center, 100) colvec<-colfunc(100)[z.facet.range] par(mar=c(2,4,1,0)) persp(y1,y2,f12,col=colvec,phi=20,theta=-60, ticktype="detailed",expand=0.6,xlab="y1",ylab="y2",zlab="f(y1,y2)",cex.axis=0.6) @ \vspace*{-1.0 in} <>= par(pty='s') y1<-y2<-seq(-3,3,by=0.01) f12<-outer(y1,y2,ffunc) image.plot(y1,y2,f12,col=colfunc(100),xlab=expression(y[1]),ylab=expression(y[2]),axes=F) axis(1, at = seq(-3, 3, by = 1),cex.axis=0.8) axis(2, at = seq(-3, 3, by = 1),cex.axis=0.8) contour(y1,y2,f12,add=T) @ \textbf{Marginal distributions:} we may compute the marginal pdfs by noting that \[ F_{Y_1}(y_1) = F_{Y_1,Y_2}(y_1,\infty) = \frac{e^{y_1}}{(1+e^{y_1})} \qquad y_1 \in \R \] and \[ F_{Y_2}(y_2) = F_{Y_1,Y_2}(\infty,y_2) = \frac{e^{y_2}}{(1+e^{y_2})} \qquad y_2 \in \R \] By differentiation, we have that \[ f_{Y_1}(y_1) =\frac{e^{y_1}}{(1+e^{y_1})^2} \qquad y_1 \in \R \] and \[ f_{Y_2}(y_2) =\frac{e^{y_2}}{(1+e^{y_2})^2} \qquad y_2 \in \R \] Here we have that for all $(y_1,y_2)$ \[ f_{Y_1,Y_2}(y_1,y_2) = f_{Y_1}(y_1) f_{Y_2}(y_2) \] so therefore the random variables $Y_1$ and $Y_2$ are \textbf{independent}. \bigskip \textbf{Conditional pdfs:} Due to independence, we have that \[ f_{Y_1|Y_2}(y_1|y_2) = f_{Y_1}(y_1) \] for all $y_1$ and \[ f_{Y_2|Y_1}(y_2|y_1) = f_{Y_2}(y_2) \] for all $y_2$. <>= par(mar=c(4,4,2,0)) y<-seq(-3,3,by=0.001) Fy<-exp(y)/(1+exp(y)) plot(y,Fy,type='l',xlab=expression(y[1]),ylab=expression(F[Y[1]](y[1])),ylim=range(0,1),cex.axis=0.8) title(expression(F[Y[1]](y[1]))) fy<-exp(y)/(1+exp(y))^2 plot(y,fy,type='l',xlab=expression(y[1]),ylab=expression(f[Y[1]](y[1])),ylim=range(0,0.5),cex.axis=0.8) title(expression(f[Y[1]](y[1]))) @ \end{document}