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Subsections

8.2 Jointly Distributed Random Variables

When the behavior of a system depends on more than one random input, some of which may be interdependent, the relationships among these random inputs need to be considered for uncertainty analysis. This is especially useful in situations where the knowledge of one variable provides information regarding other variables, thus restricting the range of possible values the other variables can assume. Such relationships can be described by joint probability distributions.

To illustrate the concept of probability analysis involving many random variables, the case for two random variables is presented first, and the results are extended for the general case involving a set of random variables. Given two random variables $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ , the probabilities

$\begin{displaymath}\ensuremath{\mbox{Pr}}\{{\ensuremath{\textstyle\mbox{\boldmat... ...uremath{F_{{\ensuremath{\textstyle\mbox{\boldmath ${y}$}}}}(y)}\end{displaymath}$

can be defined. Further, considering the event that $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ $\le x$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ $\le y$ , the joint distribution function can be defined as follows:

$\begin{displaymath}\ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}... ... x, {\ensuremath{\textstyle\mbox{\boldmath ${y}$}} } \le y \} \end{displaymath}$

The joint probability density function $\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath${xy}$ }}}}(x,y)}$ is defined as:

$\begin{displaymath}\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}... ...style\mbox{\boldmath ${xy}$}}}}(x,y)} }{\partial x \partial y} \end{displaymath}$

$\ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath${x}$ }}}}(x)}$ is called the marginal distribution function of $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ , and $\ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath${y}$ }}}}(y)}$ is the marginal distribution of $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ . Similarly, $\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath${x}$ }}}}(x)}$ is called the marginal density function of $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ , and $\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath${y}$ }}}}(y)}$ is the marginal density function of $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ . The following relationships hold for the joint distribution function:

$\textstyle \displaystyle \ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath$... ...nfty,y)} = \ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath${y}$ }}}}(y)}$		(8.3)
$\textstyle \displaystyle \ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath$... ...ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath${xy}$ }}}}(x,-\infty)} = 0$		(8.4)
$\textstyle \displaystyle\int^{\infty}_{-\infty}\ensuremath{f_{{\ensuremath{\tex... ...x,y)} dx = \ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath${y}$ }}}}(y)}$		(8.5)

Further, the probability that the random variables $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ lie in intervals given by [

] and [

] is given by:

$\begin{displaymath}\ensuremath{\mbox{Pr}}\{x_1 \le {\ensuremath{\textstyle\mbox{... ...{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}}}}(x,y)} dy dx \end{displaymath}$

In general, when

random variables are considered, the joint distribution function of random variables $\ensuremath{\textstyle\mbox{\boldmath${x_1,x_2,\ldots,x_n}$ }}$ is given by:

$\begin{displaymath}\ensuremath{\mbox{Pr}}\{ {\ensuremath{\textstyle\mbox{\boldma... ...\mbox{\boldmath ${x_1,x_2,\ldots,x_n}$}}}}(x_1,x_2,\ldots,x_n)}\end{displaymath}$

The probability density function is given by:

$\begin{displaymath}\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${x_1,x... ...\ldots,x_n)} } {\partial x_1 \partial x_2 \ldots \partial x_n} \end{displaymath}$

The marginal distribution function of random variable $\ensuremath{\textstyle\mbox{\boldmath${x_i}$ }}$ is given by:

$\begin{displaymath}\ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath ${x_i}$... ...ldots,x_n}$}}}}(\infty,\infty,\ldots,x_i,\infty,\ldots,\infty)}\end{displaymath}$

The marginal density function of $\ensuremath{\textstyle\mbox{\boldmath${x_i}$ }}$ is given by:

$\begin{displaymath}\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${x_i}$... ...x_2,\ldots,x_n)} dx_1 dx_2 \cdots dx_{i-1}dx_{i+1} \cdots dx_n \end{displaymath}$

Further, the expected value of a function $g({\ensuremath{\textstyle\mbox{\boldmath${x_1,x_2,\ldots,x_n}$ }} })$ is given by:

$\begin{displaymath}\ensuremath{\mbox{E}\{g({\ensuremath{\textstyle\mbox{\boldmath ${x_1,x_2,\ldots,x_n}$}}})\}} = \end{displaymath}$

$\begin{displaymath}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\!\!\!\!\ldots\... ..._2,\ldots,x_n}$}}}}(x_1,x_2,\ldots,x_n)} dx_1 dx_2 \ldots dx_n \end{displaymath}$

8.2.1 Moments of Jointly Distributed Random Variables

For two jointly distributed random variables $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ , the

th joint moment is defined as follows:

$\begin{displaymath}m_{i,j} = \ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{... ...{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}}}}(x,y)} dx dy \end{displaymath}$

Similarly, for the case of

random variables, $\ensuremath{\textstyle\mbox{\boldmath${x_1}$ }}$ , $\ensuremath{\textstyle\mbox{\boldmath${x_2}$ }}$ ,..., $\ensuremath{\textstyle\mbox{\boldmath${x_n}$ }}$ , the $i_1,i_2,\ldots,i_n$ th moment is defined as:

$\begin{displaymath}m_{i_1,i_2,\ldots,i_n} = \ensuremath{\mbox{E}\{{\ensuremath{\... ...ts{\ensuremath{\textstyle\mbox{\boldmath ${x_n}$}}}^{i_n}\}} = \end{displaymath}$

$\begin{displaymath}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\!\!\!\!\ldots\... ..._2,\ldots,x_n}$}}}}(x_1,x_2,\ldots,x_n)} dx_1 dx_2 \ldots dx_n \end{displaymath}$

The covariance of two random variables, widely used in probabilistic analysis, is defined as

$\begin{displaymath}\mu_{1,1} = \ensuremath{\mbox{E}\{({\ensuremath{\textstyle\mb... ...a_x)({\ensuremath{\textstyle\mbox{\boldmath ${y}$}}}-\eta_y)\}}\end{displaymath}$

The correlation coefficient of $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ is defined as

$\begin{displaymath}r_{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}} }} = \frac{\mu_{1,1}}{\sigma_x\sigma_y} \end{displaymath}$

where $\eta_x,\eta_y,\sigma_x,\sigma_y$ are means and standard deviations of $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ respectively. The covariance matrix, $\ensuremath{\textstyle\mbox{\boldmath${\Sigma}$ }}$ , of

random variables is defined as a symmetric matrix with the following elements:

$\begin{displaymath}\Sigma_{i,j} = \ensuremath{\mbox{E}\{({\ensuremath{\textstyle... ...\ensuremath{\textstyle\mbox{\boldmath ${x_j}$}}}-\eta_{x_j})\}}\end{displaymath}$

The uncertainty in model parameters are very often provided in terms of the covariance matrix.

8.2.2 Dependence of Random Variables

Two random variables $\ensuremath{\textstyle\mbox{\boldmath${x}$ }}$ and $\ensuremath{\textstyle\mbox{\boldmath${y}$ }}$ are uncorrelated if

$\begin{displaymath}\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath ... ...th{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath ${y}$}}}\}}\end{displaymath}$

They are orthogonal if

$\begin{displaymath}\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath ${x}$}}}{\ensuremath{\textstyle\mbox{\boldmath ${y}$}}}\}} = 0 \end{displaymath}$

and they are independent if

$\begin{displaymath}\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${xy}$}... ...uremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${y}$}}}}(y)}\end{displaymath}$

Similarly,

random variables, $\ensuremath{\textstyle\mbox{\boldmath${x_1,x_2,\ldots,x_n}$ }}$ , are uncorrelated if

$\begin{displaymath}\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath ... ...tstyle\mbox{\boldmath ${x_j}$}}}\}}\mbox{\ for all\ } i \ne j \end{displaymath}$

They are orthogonal if

$\begin{displaymath}\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath ... ...le\mbox{\boldmath ${x_j}$}}}\}} = 0 \mbox{\ for all\ } i \ne j \end{displaymath}$

and they are independent if

$\begin{displaymath}\ensuremath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${x_1,x... ...ath{f_{{\ensuremath{\textstyle\mbox{\boldmath ${x_n}$}}}}(x_n)}\end{displaymath}$

The interested reader can find additional material on probability concepts from the excellent texts on this subject [160,114,201,75]

Next: 9. BASIC PBPK MODEL Up: 8. PROBABILISTIC APPROACH FOR Previous: 8.1 Random Variables

Sastry S. Isukapalli
1999-01-19