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Introduction; Review of Random Variables

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Probability   ::   Set Theory   ::   Definition   ::   Conditional   ::   Random   ::   Distribution   ::   R.V.S.

Distribution functions

The cumulative distribution function (cdf) of an r.v. $X$ is defined for each $a \in \Rbb$ as

\begin{displaymath}\begin{aligned}
F_X(a) &= P(X \leq a)\\
& = P(\{\omega \in ...
...ert X(\omega) \leq a\}) \\
&= P_X((-\infty,a])
\end{aligned}\end{displaymath}

Properties of cdf:
  1. $F_X$ is non-decreasing: If $a < b$ then $F_X(a) \leq F_X(b)$ .
  2. $\lim_{a\rightarrow \infty} F_X(a) = 1$
  3. $\lim_{a \rightarrow -\infty} F_X(a) = 0.$
  4. $F_X$ is right-continuous: $\lim_{b \rightarrow a^+} F_X(b) = F_X(a)$ .
Draw "typical'' picture.






These four properties completely characterize the family of cdfs on the real line. Any function which satisfies these has a corresponding probability distribution.

  1. For $b > a$ : $P(a < X \leq b) = F_X(b) - F_X(a)$ .
  2. $P(X=a_0) = F_X(a_0) - \lim_{a\rightarrow a_0^-} F_x(a)$ . Thus, if $F_X$ is continuous at $a_0$ , $P(X=a_0) = 0$ .
From these properties, we can assign probabilities to all intervals from knowledge of the cdf. Thus we can extend this to all Borel sets.

Thus $F_X$ determines a unique probability distribution on $(\Rbb,\Bc)$ , so $F_X$ and $P_X$ are uniquely related.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Introduction; Review of 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/lec1_6.html. This work is licensed under a Creative Commons License Creative Commons License