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Change of Variable Theorems

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One Dimension  ::  Multiple Dimensions

Changing Variables: Multiple dimensions

Consider now multiple variables. Let $\gbf: \Rbb^n \rightarrow \Rbb^n$ where

\begin{displaymath}\Ybf = \gbf(\Xbf)
\end{displaymath}

We go with the equal probability idea. The probability of falling in a region in $\Xbf$ space should be the same as the probability of falling in the corresponding region in $\Ybf$ space. We'll draw pictures in two dimensions, but the concepts apply to higher dimensions.

\begin{displaymath}P(x_1 < X_1 < x_1 + dx_1, x_2 < X_2 < x_2 + dx_2) \approx
f_X(x_1,x_2)dx_1 dx_2
\end{displaymath}

Suppose the region $dx_1 dx_2$ maps to the region $dA$ in the $y$ coordinates. Equating probabilities We have

\begin{displaymath}f_Y(y_1,y_2) dA = f_X(x_1,x_2)dx_1 dx_2
\end{displaymath}

We need to evaluate $dA$ .

The region $dA$ is a parallelepiped described by the vectors

\begin{displaymath}(\frac{dy_1}{dx_1} dx_1, \frac{dy_2}{dx_1}dx_1) \qquad \text{and}
\qquad
(\frac{dy_1}{dx_2}dx_2, \frac{dy_2}{dx_2}dx_2)
\end{displaymath}

Fact: Recall from calculus that that (signed) area of the parallelepiped described by the vectors $\vbf = (v_1,v_2)$ and $\wbf = (w_1,w_2)$ is obtained from the cross product. Let us express this in matrix form

\begin{displaymath}\vbf \otimes \wbf = \det \begin{bmatrix}\ibf & \jbf &\kbf \\
v_1 & v_2 & 0 \\
w_1 & w_2 & 0
\end{bmatrix}\end{displaymath}

In our case, we have

\begin{displaymath}\begin{aligned}
\det \begin{bmatrix}
\ibf & \jbf & \kbf \\
...
..._2 - \frac{dy_2}{dx_1}dx_1
\frac{dy_1}{dx_2}dx_2)
\end{aligned}\end{displaymath}

The (signed) area is then

\begin{displaymath}\frac{dy_1}{dx_1}dx_1 \frac{dy_2}{dx_2}dx_2 - \frac{dy_2}{dx_...
...c{dy_2}{dx_2} -
\frac{dy_2}{dx_1}d\frac{dy_1}{dx_2}) dx_1 dx_2
\end{displaymath}

Let

\begin{displaymath}J = \det \begin{bmatrix}
\frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\
\frac{dy_2}{dx_1} & \frac{dy_2}{dx_2}
\end{bmatrix}\end{displaymath}

This matrix of partial derivatives is called the Jacobian of the function $\gbf$ .

Back to probabilities. We have

\begin{displaymath}f_Y(y_1,y_2) dA = f_X(x_1,x_2)dx_1 dx_2
\end{displaymath}

or

\begin{displaymath}f_Y(y_1,y_2) \vert J\vert dx_1 dx_2 = f_X(x_1,x_2) dx_1 dx_2
\end{displaymath}

or

\begin{displaymath}f_Y(y_1,y_2) = \vert J\vert^{-1} f_X(x_1,x_2)
\end{displaymath}

or, in general for an invertible function $\gbf$ ,

\begin{displaymath}\boxed{f_Y(\ybf) = \vert J\vert^{-1} f_X(\gbf^{-1}(\ybf))}
\end{displaymath}


\begin{example}
Box-Muller transformation. Let $X_1 \sim \Uc(0,1)$ and $X_2 \s...
...Y_1\sim \Nc(0,1) \qquad \qquad Y_2 \sim \Nc(0,1).
\end{displaymath}\end{example}

Many-to-one mappings

Let $Y = g(X_1,X_2)$ where $X_1$ and $X_2$ are jointly distributed random variables. For a given value of $y$ , the inverse may form a curve in $(x_1,x_2)$ space. Let $A_y$ denote the region in the $X_1X_2$ plane such that $g(X_1,X_2) \leq y$ . (This may not be a connected region.) Then

\begin{displaymath}\{Y \leq y \} = \{g(X_1,X_2) \leq y\} = \{ (X_1,X_2) \in A_y\}
\end{displaymath}

so

\begin{displaymath}F_Y(y) = \int\int_{A_y} f_{X_1X_2}(x_1,x_2)dx_1dx_2
\end{displaymath}

Let $\Delta A_y$ denote the region of the $X_1X_2$ plane such that $y <
g(x_1,x_2) \leq y+dy$ . then

\begin{displaymath}f_Y(y) dy = \int\int_{\Delta A_y} f_{X_1X_2}(x_1,x_2)dx_1dx_2
\end{displaymath}


\begin{example}
Let $Z = X+Y$. The region in the $xy$ plane such that $x+y \le...
...-\infty}^\infty f_{XY}(z-y,y) dy.
\end{displaymath}Independence...
\end{example}

\begin{example}
Let $Z = X/Y$. The region of the plane such that $x/y \leq z$ ...
...int_{-\infty}^\infty f_{XY}(zy,y)\vert y\vert dy.
\end{displaymath}\end{example}

\begin{example}
Let $Z = \sqrt{X^2 + Y^2}$. The region $A_z$ is the circle $x^...
...nd{displaymath}$z> 0$. This is a {\em Rayleigh} distribution.
\par
\end{example}

Sometimes it is helpful to introduce an auxiliary variable, then integrate it out.

Suppose $Z = XY$ . Introduce the auxiliary variable $W = X$ . Then the inverse functions are straightforward to compute,

\begin{displaymath}X = W \text{ and } Y = Z/W.
\end{displaymath}

The Jacobian is

\begin{displaymath}J = \begin{bmatrix}\partiald{z}{x} & \partiald{z}{y} \\
\partiald{w}{x} & \partiald{w}{y}
\end{bmatrix} = -x = -w.
\end{displaymath}

The joint density is

\begin{displaymath}f_{ZW}(z,w) = \frac{1}{\vert w\vert} f_{XY}(w,z/w).
\end{displaymath}

Then the density of $Z$ can be obtained by integration:

\begin{displaymath}f_Z(z) = \int_{-\infty}^\infty \frac{1}{\vert w\vert} f_{XY}(w,z/w) dw.
\end{displaymath}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Change of Variable Theorems. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec4_2.html. This work is licensed under a Creative Commons License Creative Commons License