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More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

Pairs of random variables

Ultimately, we will be dealing with infinite sequences of random variables. As steps along the way, we will examine carefully pairs of random variables, then vectors of random variables.

On $\Rbb^2$ , the smallest $\sigma$ -field of interest is $\Bc^2$ , which is the smallest $\sigma$ -field containing all of the rectangles. This is the Borel $\sigma$ -field of $\Rbb^2$ .
\begin{definition}
A {\bf bivariate random variable} $(X,Y)$ is a measurable mapping
from $(\Omega,\Fc)$ to $(\Rbb^2,\Bc^2)$.
\end{definition}
That is,

\begin{displaymath}\{\omega \in \Omega : (X,Y)(\omega) \in B\} \in \Fc \forall B \in
\Bc^2
\end{displaymath}

Note that two r.v.s $X, Y$ on $(\Omega,\Fc)$ form a bivariate r.v.


\begin{definition}
The {\bf joint} or {\bf bivariate} distribution of $(X,Y)$ ...
...ga: (X,Y)(\omega) \in B\})
\end{displaymath}for $B \in \Bc^2$.
\end{definition}

\begin{definition}
The {\bf joint c.d.f.} of $(X,Y)$ is defined as
\begin{disp...
...b} = \{(x,y) \in \Rbb^2: x \leq a, y \leq b\}.
\end{displaymath}\end{definition}
Properties of the joint c.d.f.:

  1. $\lim_{a,b \rightarrow \infty} F_{X,Y}(a,b) = 1$ .
  2. $\lim_{a \rightarrow -\infty} F_{X,Y}(a,b) = 0 = \lim_{b
\rightarrow - \infty} F_{X,Y}(a,b)$ .
  3. $\lim_{a\rightarrow \infty} F_{X,Y}(a,b) = F_Y(b)$ , the marginal c.d.f. of $Y$ .

    $\lim_{b\rightarrow \infty} F_{X,Y}(a,b) = F_X(a)$ , the marginal c.d.f. of $X$ .

  4. $F_{XY}(a,b)$ is continuous ''from the northeast.''
  5. $F_{XY}(x,y)$ is montonically increasing (or, more precisely, nondecreasing) in both variables.
Any function with these properties is a legitimate c.d.f., and completely characterizes the family of joint c.d.f.s.

Joint discrete r.v.s


\begin{definition}
If $X,Y$ are discrete r.v.s taking values in sets $\{x_1,\l...
...by
\begin{displaymath}p_{XY}(a,b) = P(X=a,Y=b)
\end{displaymath}\end{definition}
Properties of $p_{XY}$ :

  1. $p_{XY} \geq 0$ , and $p_{XY}(a,b) = 0$ if $a \not\in \{x_1,\ldots\}$ or $b \not \in \{y_1,\ldots\}$
  2. $\sum_{i=1}^\infty \sum_{j=1}^\infty p_{XY}(x_i,y_j) = 1$ .
  3. $F_{XY}(a,b) = \sum_{\{x_i,y_i\}: x_i \leq a, y_j \leq b\}}
p_{XY}(x_i,y_j) $
  4. Marginals:

    \begin{displaymath}P_X(x_i) = \sum_{j} p_{XY}(x_i,y_j)
\end{displaymath}


    \begin{displaymath}P_Y(y_j) = \sum_{i} p_{XY}(x_i,y_j)
\end{displaymath}

Joint continuous r.v.s


\begin{definition}
$X$ and $Y$ are {\bf jointly continuous} r.v.s if there is...
...alled the {\bf joint p.d.f.} of $X$ and $Y$ (when it exists).
\end{definition}
Properties of joint p.d.f.:

  1. $f_{XY} \geq 0$ .
  2. $\int_{-\infty}^\infty\int_{-\infty}^\infty f_{XY}(x,y) dx,dy =
1$ .
  3. We can get the p.d.f. from the c.d.f:

    \begin{displaymath}f_{XY}(x,y) = \frac{\partial^2}{\partial x \partial y} F_{XY}(x,y).
\end{displaymath}

  4. Marginals:

    \begin{displaymath}f_X(x) = \int_{-\infty}^\infty f_{XY}(x,y) dy
\end{displaymath}


    \begin{displaymath}f_Y(y) = \int_{-\infty}^\infty f_{XY}(x,y) dx
\end{displaymath}

    If $X$ and $J$ are jointly continuous, then they are marginally continuous (that is, $f_X$ and $f_Y$ are continuous). However, the opposite is not true.

\begin{example}
Important example! $X$ and $Y$ are said to be {\bf jointly
G...
... coefficient. It is a measure of how
much $X$ \lq\lq looks like'' $Y$.
\end{example}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec2_3.html. This work is licensed under a Creative Commons License Creative Commons License