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Introduction; Review of Random Variables

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

Pure types of r.v.s

  1. Discrete r.v.s -- an r.v. whose possible values can be enumerated.
  2. Continuous r.v.s -- an r.v. whose distribution function can be written as the (regular) integral of another function
  3. Singular, but not discrete -- Any other r.v.

Discrete r.v.s

A random variable that can take on at most a countable number of possible values is said to be a discrete r.v.:

\begin{displaymath}X: \Omega \rightarrow \{x_1,x_2,\ldots\}
\end{displaymath}


\begin{definition}
For a discrete r.v. $X$, we define the probability mass func...
...here $p_X(a) = 0$ if $a \neq x_i$ for any r.v. outcome $x_i$.
\end{definition}
Properties of pmfs:
  1. Nonnegativity:

    \begin{displaymath}p_X(a) =
\begin{cases}
\geq 0 & a \in \{x_1,x_2,\ldots\} \\
0 & \text{else}
\end{cases}\end{displaymath}

  2. Total probability:

    \begin{displaymath}\sum_{i=1}^\infty p_X(x_i) = 1.
\end{displaymath}

    (These two properties completely characterize the class of all pdfs on a given set $\{x_1,x_2,\ldots\}$ .)
  3. Relation to cdf:

    \begin{displaymath}p_X(a) = F_X(a) - \lim_{b \rightarrow a^-} F_X(b)
\end{displaymath}

    (To the pdf and the cdf contain the same information. Note that for a discrete r.v., the cdf is piecewise constant. Draw picture)
  4. $F_X(a)= \sum_{\{x_i\vert x_i\leq a\}} p_X(x_i)$


\begin{example}
{\bf Bernoulli} with parameter $\pi$, ($0 \leq \pi \leq 1$).
\...
...i is often used to model bits, bit errors, random coin
flips, etc.
\end{example}

\begin{example}
{\bf Binomial} $(n,\pi)$, ($0 \leq \pi \leq 1, n \in \Zbb^+$).
...
...^k b^{n-k}}
\end{displaymath}\par (Plot the probability function.)
\end{example}

\begin{example}
{\bf Poisson} $\lambda$:
\begin{displaymath}X : \Omega \rightar...
...e this expression.
\par
How do we find $\sum_{k=0}^\infty P(X=k)$?
\end{example}

Continuous r.v.s

A r.v. $X$ is said to be continuous is there is a function $f_X: \Rbb
\rightarrow \Rbb$ such that

\begin{displaymath}F_X(a) = \int_{-\infty}^a f_X(x) dx
\end{displaymath}

for all $a \in \Rbb$ . In this case, $F_X(a)$ is an absolutely continuous function. 1

Properties:

  1. $f_X(x) \geq 0$
  2. $\int_{-\infty}^\infty f_X(x) dx = 1$ .
These two properties completely characterize the $f_X$ s.

$f_X$ called the probability density function (pdf) of $X$ .

  1. $P(X \in B) = \int_{B} f_X(x) dx = P_X(B), B \in \Bc$ .
  2. $P(X=a) = 0$ for all $a \in \Rbb$ .
  3. $P(X \in [x_0,x_0 + \Delta x] \approx f(x_0) \Delta x$ .

    \begin{displaymath}P(X \in dx) = P_X(dx) = dP = f(x)dx.
\end{displaymath}


\begin{example}
{\bf Uniform} on $(\alpha,\beta)$ (with $\beta > \alpha$):
\be...
...lot pdf and cdf.
\par Uses: \lq\lq random'' number. Phase distribution.
\end{example}

\begin{example}
{\bf Exponential}$(\lambda)$. ($\lambda > 0$).
\begin{displayma...
...
\par
Uses: Waiting time. (We'll see later what we mean by this.)
\end{example}

\begin{example}
Gaussian (normal) $N(\mu,\sigma^2)$:
\begin{displaymath}\boxed{...
...{1}{\sqrt{2\pi}}\int_{x}^{\infty}
e^{-z^2/2}\, dz
\end{displaymath}\end{example}

Properties of Gaussian Random Variables

Let $X \sim N(\mu,\sigma^2)$ .

  1. If $Y = \alpha X + \beta$ then $Y \sim N(\alpha\mu+\beta,
\alpha^2\sigma^2)$
  2. $(X-\mu)/\sigma \sim N(0,1)$
  3. $F_X(a) = \Phi((a-\mu)/\sigma)$ .
  4. $\Phi(-x) = 1-\Phi(x)$ .
  5. A Gaussian r.v. is completely characterized by its first two moments (i.e., the mean and variance).
We will also see, and make use of, many other properties of Gaussian random processes, such as the fact that an uncorrelated Gaussian r.p. is also independent.

Bibliography

  1. P. Billingsley, Probability and Measure . New York: Wiley, 1986.

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_7.html. This work is licensed under a Creative Commons License Creative Commons License