ECE600 F13 Functions of Two Random Variables mhossain - Rhea

Random Variables and Signals

Topic 13: Functions of Two Random Variables

One Function of Two Random Variables

Given random variables X and Y and a function g:R $$ ^2 $$ →R, let Z = g(X,Y). What is f $$ _Z $$ (z)?

We assume that Z is a valid random variable, so that ∀z ∈ R, there is a D $$ _z $$ ∈ b(R $$ ^2 $$ ) such that

\{Z\leq z\} = \{(X,Y)\in D_z\}

i.e.

D_z = \{(x,y)\in\mathbb R^2:g(x,y)\leq z\}

Then,

F_Z(z) = P((X,Y)\in D_z)=\int\int_{D_z}f_{XY}(x,y)dxdy

and we can find f $$ _Z $$ from this.

Example $\qquad$ g(x,y) = x + y. So Z = X + Y. Here,

D_z = \{(x,y)\in\mathbf R^2:x+y\leq z\}

and

F_Z(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{z-x}f_{XY}(x,y)dydx

Fig 1: The shaded region indicates D

$ _z $

.

If we now assume that X and Y are independent, then

\begin{align} F_Z(z) &= \int_{-\infty}^{\infty}\int_{-\infty}^{z-x}f_X(x)f_Y(y)dydx \\ &=\int_{-\infty}^{\infty}f_X(x)F_Y(z-x)dx \end{align}

Then,

\begin{align} f_Z(z) &= \frac{d}{dz}[\int_{-\infty}^{\infty}f_X(x)F_Y(z-x)dx] \\ &=\int_{-\infty}^{\infty}f_X(x)\frac{dF_Y(z-x)}{dz}dx \\ &=\int_{-\infty}^{\infty}f_X(x)f_Y(z-x)dx \\ &=(f_X\ast f_Y)(z) \end{align}

So if X and Y are independent, and Z = X + Y, we can find f $$ _Z $$ by convolving f $$ _X $$ and f $$ _Y $$ .

Example $\qquad$ X and Y are independent exponential random variables with means $\mu_X$ = $\mu_Y$ = $\mu$ . So,

\begin{align} f_Z(z)&=\int_{-\infty}^{\infty}\frac{1}{\mu}e^{-\frac{x}{\mu}}u(x)\frac{1}{\mu}e^{-\frac{(z-x}{\mu}}u(z-x)dx \\ &=\int_0^z\frac{1}{\mu^2}e^{-\frac{z}{\mu}}dx\;\;\mbox{for}\;z=0 \\ &=\frac{z}{\mu^2}e^{-\frac{z}{\mu}}u(z) \end{align}

So the sum of two independent exponential random variables is not an exponential random variable.

Given two random variables X and Y, cobsider random variables Z and W defined as follows:

\begin{align} Z &= g(X,Y) \\ W &= h(X,Y) \end{align}

where g:R $$ ^2 $$ →R and h:R $$ ^2 $$ →R.

For example, if we have a linear transformation, then

\begin{bmatrix}Z\\W\end{bmatrix} =A \begin{bmatrix}X\\Y\end{bmatrix}

where A is a 2x2 matrix. In this case,

\begin{align} g(x,y) &= a_{11}x+a_{12}y \\ h(x,y) &= a_{21}x+a_{22}y \end{align}

where

A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

How do we find f $_{ZW}$ ?

If we could find D $_{z,w}$ = {(x,y) ∈ R $$ ^2 $$ : g(x,y) ≤ z and h(x,y) ≤ w} for every (z,w) ∈ R $$ ^2 $$ , then we could compute

\begin{align} F_{ZW}(z,w) &= p(Z\leq z, W\leq w) \\ &=P((X,Y)\in D_{z,w}) \\ &=\int\int_{D_{z,w}}f_{XY}(x,y)dxdy \end{align}

It is often quite difficult to find D $_{z,w}$ , so we use a formula for f $_{ZW}$ instead.

Assume that the functions g and h satisfy

z = g(x,y) and w = h(x,y) can be solved simultaneously for unique x and y. We will write x = g $^{-1}$ (z,w) and y = h $^{-1}$ (z,w) where g $^{-1}$ :R $$ ^2 $$ →R and h $^{-1}$ :R $$ ^2 $$ →R.

For the linear transformation example, this means we assume A

^{-1}

exists (i.e. it is invertible). In this case

\begin{align} g^{-1}(z,w) &= b_{11}z+b_{12}w \\ h^{-1}(z,w) &= b_{21}z+b_{22}w \end{align}

where

A^{-1}=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

\frac{\partial x}{\partial z},\;\frac{\partial x}{\partial w},\;\frac{\partial y}{\partial z},\;\frac{\partial y}{\partial w},\;\frac{\partial z}{\partial x},\;\frac{\partial z}{\partial y},\;\frac{\partial w}{\partial x},\;\frac{\partial w}{\partial y}

exist.

Then it can be shown that

f_{ZW}(z,w) = \frac{f_{XY}(g^{-1}(z,w),h^{-1}(z,w))}{\left|\frac{\partial\left(z,w\right)}{\partial\left(x,y\right)}\right|}

where

\begin{align} \left|\frac{\partial(z,w)}{\partial(x,y)}\right| &\equiv \left|\begin{bmatrix} \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} \end{bmatrix}\right| \\ &= \left|\frac{\partial z}{\partial x}\frac{\partial w}{\partial y}-\frac{\partial z}{\partial y}\frac{\partial w}{\partial x} \right| \end{align}

z=g(x,y)\qquad x=h(x,y)

Note that we take the absolute value of the determinant. This is called the Jacobian of the transformation. For more on the Jacobian, see here.

The proof for the above formula is in Papoulis.

Example