Review of probability theory

The focus of this course is on digital communication, which involves transmission of information, in its most general sense,from source to destination using digital technology. Engineering such a system requires modeling both the informationand the transmission media. Interestingly, modeling both digital or analog information and many physical media requires aprobabilistic setting. In this chapter and in the next one we will review the theory of probability, model random signals, andcharacterize their behavior as they traverse through deterministic systems disturbed by noise and interference. Inorder to develop practical models for random phenomena we start with carrying out a random experiment. We then introducedefinitions, rules, and axioms for modeling within the context of the experiment. The outcome of a random experiment isdenoted by $$ . The sample space $$ is the set of all possible outcomes of a random experiment. Such outcomescould be an abstract description in words. A scientific experiment should indeed be repeatable where each outcome couldnaturally have an associated probability of occurrence. This is defined formally as the ratio of the number of times the outcomeoccurs to the total number of times the experiment is repeated.

Random variables

A random variable is the assignment of a real number to each outcome of a random experiment.

Distributions

Probability assignments on intervals $a< X\le b$

Cumulative distribution

The cumulative distribution function of a random variable $X$ is a function $F(X, (\mathbb{R}, \mathbb{R}))$ such that

$$F(X, b)=(X\le b)=(\{\in \colon X()\le b\})$$

Continuous Random Variable

A random variable $X$ is continuous if the cumulative distribution function can bewritten in an integral form, or

$$F(X, b)=\int_{()} \,d x$$∞ b f X x

and $f(X, x)$ is the probability density function (pdf) ( e.g. , $F(X, x)$ is differentiable and $f(X, x)=\frac{d F(X, x)}{d x}$ )

Discrete Random Variable

A random variable $X$ is discrete if it only takes at most countably many points( i.e. , $F(X, )$ is piecewise constant). The probability mass function (pmf) is defined as

$$p(X, x_k)=(X=x_k)=F(X, x_k)-\lim_{x\to x_k\land (x< x_k)}F(X, x)$$

Two random variables defined on an experiment have joint distribution

Joint pdf can be obtained if they are jointly continuous

Joint pmf if they are jointly discrete

Conditional density function

Two random variables are independent if

Moments

Statistical quantities to represent some of the characteristics of a random variable.

• Mean
  $${}_X=\langle X\rangle$$
• Second moment
  $$(X^2)=\langle X^2\rangle$$
• Variance
  $$\mathrm{Var}(X)=\sigma(X)^2=\langle (X-{}_X)^2\rangle =\langle X^2\rangle -{}_X^2$$
• Characteristic function
  $${}_X(u)=\langle e^{iuX}\rangle$$
  for $u\in \mathbb{R}$ , where $i=\sqrt{-1}$
• Correlation between two random variables
  $$R_{XY}=\langle X{Y}^{*}\rangle =\begin{cases}\int_{()} \,d y & \text{if $$}\end{cases}$$∞ ∞ x ∞ ∞ x y * f X Y x y X and Y are jointly continuous k k l l x k y l * p X Y x k y l X and Y are jointly discrete
• Covariance
  $$C_{XY}=\mathrm{Cov}(X, Y)=\langle (X-{}_X)(Y-{}_Y)^{*}\rangle =R_{XY}-{}_X{}_Y^{*}$$
• Correlation coefficient
  $${}_{XY}=\frac{\mathrm{Cov}(X, Y)}{{}_X{}_Y}$$

Uncorrelated random variables

Two random variables $X$ and $Y$ are uncorrelated if ${}_{XY}=0$ .