Digital Modulation Basics 2.10 2001/06/01 2004/01/07 09:54:51 US/Central Behnaam Aazhang aaz@ece.rice.edu Roy Ha rha@rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Behnaam Aazhang aaz@ece.rice.edu blargh blargh The outcome of a random experiment is denoted by ω. The sample space Ω is the set of all possible outcomes of a random experiment.

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

Roll a dice. Outcomes ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω i i dots on the face of the dice. X ω i i X i X ω i i ω Ω

Distributions Probability assignments on intervals a X b Cumulative distribution The cumulative distribution function of a random variable X is a function F X such that → ℝ ℝ such that F X b X b X ω b ω Ω

Continuous Random Variable A random variable X is continuous if F X b x b f X x and f X x is the probability density function (pdr) (e.g., F X x is differentiable and f X x x F X 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 x x x k x x k F X x Two random variables defined on an experiment have joint distribution F X Y a b X a Y b X ω a Y ω b ω Ω

Joint pdf can be obtained if they are jointly continuous. F X Y a b y b x a f X Y x y (e.g., f X Y x y x y F X Y x y ) joint pmf if they are jointly discrete p X Y x k y l X x k Y y l Conditional density function f Y | X y | x f X Y x y f X x for all x and with f X x 0