Multivariable Version Change of Variables Formula for Continuous Random Variable



5.2.4 Functions of Two Continuous Random Variables

So far, we have seen several examples involving functions of random variables. When we have two continuous random variables $g(X,Y)$, the ideas are still the same. First, if we are just interested in $E[g(X,Y)]$, we can use LOTUS:

LOTUS for two continuous random variables:

\begin{align}\label{eq:LOTUS-2D-Cont} E[g(X,Y)]=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x,y)f_{XY}(x,y) \hspace{5pt} dxdy \hspace{40pt} (5.19) \end{align}


Example
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} x+y & \quad 0 \leq x,y \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Find $E[XY^2]$.

  • Solution
    • We have \begin{align} \nonumber E[XY^2]&=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (xy^2)f_{XY}(x,y) \hspace{5pt} dxdy\\ \nonumber &=\int_{0}^{1} \int_{0}^{1} xy^2(x+y) \hspace{5pt} dxdy\\ \nonumber &=\int_{0}^{1} \int_{0}^{1} x^2y^2+xy^3 \hspace{5pt} dxdy\\ \nonumber &=\int_{0}^{1} \left(\frac{1}{3}y^2+\frac{1}{2}y^3\right) dy\\ \nonumber &=\frac{17}{72}. \end{align}

If $Z=g(X,Y)$ and we are interested in its distribution, we can start by writing \begin{align} \nonumber F_Z(z)&=P(Z \leq z)\\ \nonumber &=P(g(X,Y) \leq z)\\ \nonumber &=\iint\limits_D f_{XY}(x,y) \hspace{5pt} dxdy, \end{align} where $D=\{(x,y)|g(x,y)<z\}$. To find the PDF of $Z$, we differentiate $F_Z(z)$.


Example
Let $X$ and $Y$ be two independent $Uniform(0,1)$ random variables, and $Z=XY$. Find the CDF and PDF of $Z$.

  • Solution
    • First note that $R_Z=[0,1]$. Thus, \begin{align} \nonumber F_Z(z)&=0, \hspace{30pt} \textrm{ for }z \leq 0,\\ \nonumber F_Z(z)&=1, \hspace{30pt} \textrm{ for }z \geq 1. \end{align} For $0<z<1$, we have \begin{align} \nonumber F_Z(z)&=P(Z \leq z)\\ \nonumber &=P(XY \leq z)\\ \nonumber &=P\left(X \leq \frac{z}{Y}\right). \end{align} Just to get some practice, we will show you two ways to calculate $P(X \leq \frac{z}{Y})$ for $0<z<1$. The first way is just integrating $f_{XY}(x,y)$ in the region $x \leq \frac{z}{y}$. We have \begin{align} \nonumber P\left(X \leq \frac{z}{Y}\right)&=\int_{0}^{1} \int_{0}^{\frac{z}{y}}f_{XY}(x,y) \hspace{5pt} dxdy\\ \nonumber &=\int_{0}^{1} \int_{0}^{\min(1,\frac{z}{y})} 1 \hspace{5pt} dxdy\\ \nonumber &=\int_{0}^{1} \min\left(1,\frac{z}{y}\right) \hspace{5pt} dy. \end{align} Note that if we let $g(y)=\min\left(1,\frac{z}{y}\right)$, then \begin{equation} \nonumber g(y) = \left\{ \begin{array}{l l} 1 & \quad \textrm{for } 0<y<z\\ & \quad \\ \frac{z}{y} & \quad \textrm{for } z \leq y\leq 1 \end{array} \right. \end{equation} Therefore, \begin{align} \nonumber P\left(X \leq \frac{z}{Y}\right)&=\int_{0}^{1} g(y) \hspace{5pt} dy\\ \nonumber &=\int_{0}^{z} 1 \hspace{5pt} dy+ \int_{z}^{1} \frac{z}{y} \hspace{5pt} dy\\ \nonumber &=z-z \ln z. \end{align} The second way to find $P(X \leq \frac{z}{Y})$ is to use the law of total probability. We have \begin{align} \nonumber P(X \leq \frac{z}{Y})&=\int_{0}^{1} P(X \leq \frac{z}{Y}|Y=y) f_Y(y) \hspace{5pt} dy\\ \nonumber &=\int_{0}^{1} P\left(X \leq \frac{z}{y}\right) f_Y(y) \hspace{5pt} dy &\textrm{ (since $X$ and $Y$ are independent)} \hspace{20pt} (5.20) \end{align} Note that \begin{equation} \nonumber P\left(X \leq \frac{z}{y}\right) = \left\{ \begin{array}{l l} 1 & \quad \textrm{for } 0<y<z\\ & \quad \\ \frac{z}{y} & \quad \textrm{for } z \leq y\leq 1 \end{array} \right. \end{equation} Therefore, \begin{align} \nonumber P\left(X \leq \frac{z}{Y}\right)&=\int_{0}^{1} P\left(X \leq \frac{z}{y}\right) f_Y(y) \hspace{5pt} dy\\ \nonumber &=\int_{0}^{z} 1 \hspace{5pt} dy+ \int_{z}^{1} \frac{z}{y} \hspace{5pt} dy\\ \nonumber &=z-z \ln z. \end{align} Thus, in the end we obtain \begin{equation} \nonumber F_Z(z) = \left\{ \begin{array}{l l} 0 & \quad z \leq 0 \\ z-z \ln z & \quad 0<z<1 \\ 1 & \quad z \geq 1 \end{array} \right. \end{equation} You can check that $F_Z(z)$ is a continuous function. To find the PDF, we differentiate the CDF. We have \begin{equation} \nonumber f_Z(z) = \left\{ \begin{array}{l l} - \ln z & \quad 0<z<1 \\ 0 & \quad \textrm{otherwise} \end{array} \right. \end{equation}


The Method of Transformations:

When we have functions of two or more jointly continuous random variables, we may be able to use a method similar to Theorems 4.1 and 4.2 to find the resulting PDFs. In particular, we can state the following theorem. While the statement of the theorem might look a little confusing, its application is quite straightforward and we will see a few examples to illustrate the methodology.

Theorem
Let $X$ and $Y$ be two jointly continuous random variables. Let $(Z,W)=g(X,Y)=(g_1(X,Y),g_2(X,Y))$, where $g:\mathbb{R}^2 \mapsto \mathbb{R}^2$ is a continuous one-to-one (invertible) function with continuous partial derivatives. Let $h=g^{-1}$, i.e., $(X,Y)=h(Z,W)=(h_1(Z,W),h_2(Z,W))$. Then $Z$ and $W$ are jointly continuous and their joint PDF, $f_{ZW}(z,w)$, for $(z,w) \in R_{ZW}$ is given by \begin{align} \nonumber f_{ZW}(z,w)=f_{XY}(h_1(z,w),h_2(z,w)) |J|, \end{align} where $J$ is the Jacobian of $h$ defined by \begin{align} \nonumber J= \det \begin{bmatrix} \frac{\partial h_1}{\partial z} & \frac{\partial h_1}{\partial w} \\ & \\ \frac{\partial h_2}{\partial z} & \frac{\partial h_2}{\partial w} \\ \end{bmatrix} =\frac{\partial h_1}{\partial z}.\frac{\partial h_2}{\partial w}-\frac{\partial h_2}{\partial z}\frac{\partial h_1}{\partial w}. \end{align}

The following examples show how to apply the above theorem.


Example
Let $X$ and $Y$ be two independent standard normal random variables. Let also \begin{equation} \nonumber \left\{ \begin{array}{l} Z=2X-Y \\ W=-X+Y \end{array} \right. \end{equation} Find $f_{ZW}(z,w)$.

  • Solution
    • $X$ and $Y$ are jointly continuous and their joint PDF is given by \begin{align}%\label{} \nonumber f_{XY}(x,y)=f_X(x)f_Y(y)=\frac{1}{2 \pi} \exp\left\{-\frac{x^2+y^2}{2}\right\}, \hspace{30pt} \textrm{ for all }x,y \in \mathbb{R}. \end{align} Here, the function $g$ is defined by $(z,w)=g(x,y)=(g_1(x,y),g_2(x,y))=(2x-y,-x+y)$. Solving for $x$ and $y$, we obtain the inverse function $h$: \begin{equation} \nonumber \left\{ \begin{array}{l} x=z+w=h_1(z,w) \\ y=z+2w=h_2(z,w) \end{array} \right. \end{equation} We have \begin{align} \nonumber f_{ZW}(z,w)&=f_{XY}(h_1(z,w),h_2(z,w)) |J|\\ \nonumber &=f_{XY}(z+w,z+2w) |J|, \end{align} where \begin{align} \nonumber J= \det \begin{bmatrix} \frac{\partial h_1}{\partial z} & \frac{\partial h_1}{\partial w} \\ & \\ \frac{\partial h_2}{\partial z} & \frac{\partial h_2}{\partial w} \\ \end{bmatrix} =\det \begin{bmatrix} 1 & 1 \\ & \\ 1 & 2 \\ \end{bmatrix} =1. \end{align} Thus, we conclude that \begin{align} \nonumber f_{ZW}(z,w)&=f_{XY}(z+w,z+2w) |J|\\ \nonumber &=\frac{1}{2 \pi} \exp\left\{-\frac{(z+w)^2+(z+2w)^2}{2}\right\} \\ \nonumber &=\frac{1}{2 \pi} \exp\left\{-\frac{2z^2+5w^2+6zw}{2}\right\}. \end{align}


Example
Let $X$ and $Y$ be two random variables with joint PDF $f_{XY}(x,y)$. Let $Z=X+Y$. Find $f_{Z}(z)$.

  • Solution
    • To apply Theorem 5.1, we need two random variables $Z$ and $W$. We can simply define $W=X$. Thus, the function $g$ is given by \begin{equation} \nonumber \left\{ \begin{array}{l} z=x+y \\ w=x \end{array} \right. \end{equation} Then, we can find the inverse transform: \begin{equation} \nonumber \left\{ \begin{array}{l} x=w \\ y=z-w \end{array} \right. \end{equation} Then, we have \begin{align} \nonumber |J|= | \det \begin{bmatrix} 0 & 1 \\ & \\ 1 & -1 \\ \end{bmatrix}| =|-1|=1. \end{align} Thus, \begin{align} \nonumber f_{ZW}(z,w)=f_{XY}(w,z-w). \end{align} But since we are interested in the marginal PDF, $f_Z(z)$, we have \begin{align} \nonumber f_Z(z)=\int_{-\infty}^{\infty} f_{XY}(w,z-w)dw. \end{align} Note that, if $X$ and $Y$ are independent, then $f_{XY}(x,y)=f_X(x)f_Y(y)$ and we conclude that \begin{align} \nonumber f_Z(z)=\int_{-\infty}^{\infty} f_{X}(w)f_Y(z-w)dw. \end{align}

The above integral is called the convolution of $f_X$ and $f_Y$, and we write \begin{align} \nonumber f_Z(z)&=f_X(z)\ast f_Y(z)\\ \nonumber &=\int_{-\infty}^{\infty} f_{X}(w)f_Y(z-w)dw=\int_{-\infty}^{\infty} f_{Y}(w)f_X(z-w)dw. \end{align}

If $X$ and $Y$ are two jointly continuous random variables and $Z=X+Y$, then \begin{align} \nonumber f_Z(z)=\int_{-\infty}^{\infty} f_{XY}(w,z-w)dw=\int_{-\infty}^{\infty} f_{XY}(z-w,w)dw. \end{align} If $X$ and $Y$ are also independent, then \begin{align} \nonumber f_Z(z)&=f_X(z)\ast f_Y(z)\\ \nonumber &=\int_{-\infty}^{\infty} f_{X}(w)f_Y(z-w)dw=\int_{-\infty}^{\infty} f_{Y}(w)f_X(z-w)dw. \end{align}


Example
Let $X$ and $Y$ be two independent standard normal random variables, and let $Z=X+Y$. Find the PDF of $Z$.

  • Solution
    • We have \begin{align} \nonumber f_Z(z)&=f_X(z)\ast f_Y(z)\\ \nonumber &=\int_{-\infty}^{\infty} f_{X}(w)f_Y(z-w)dw\\ \nonumber &=\int_{-\infty}^{\infty} \frac{1}{2 \pi}e^{-\frac{w^2}{2}}e^{-\frac{(z-w)^2}{2}}dw\\ \nonumber &=\frac{1}{\sqrt{4 \pi}} e^{\frac{-z^2}{4}}\int_{-\infty}^{\infty} \frac{1}{\sqrt{\pi}}e^{-(w-\frac{z}{2})^2}dw\\ \nonumber &=\frac{1}{\sqrt{4 \pi}} e^{\frac{-z^2}{4}}, \end{align} where $\int_{-\infty}^{\infty} \frac{1}{\sqrt{\pi}}e^{-(w-\frac{z}{2})^2}dw=1$ because it is the integral of the PDF of a normal random variable with mean $\frac{z}{2}$ and variance $\frac{1}{2}$. Thus, we conclude that $Z \sim N(0,2)$. In fact, this is one of the interesting properties of the normal distribution: the sum of two independent normal random variables is also normal. In particular, similar to our calculation above, we can show the following:

Theorem
If $X \sim N(\mu_X,\sigma^2_X)$ and $Y \sim N(\mu_Y,\sigma^2_Y)$ are independent, then \begin{align}%\label{} \nonumber X+Y \hspace{5pt} \sim \hspace{5pt} N\bigg(\mu_X+\mu_Y, \sigma_X^2+\sigma_Y^2\bigg). \end{align}

We will see an easier proof of Theorem 5.2 when we discuss moment generating functions.

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