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# Digital Modulation Basics

Module by: Behnaam Aazhang. E-mail the author

Summary: blargh

The outcome of a random experiment is denoted by ωω. The sample space ΩΩ 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.

### Example 1

Roll a dice. Outcomes ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6

ω i =i ω i i dots on the face of the dice.

X ω i =i X ω i i

PrX=i=PrωΩ| X ω i =i X i X ω i i ω Ω

## Distributions

Probability assignments on intervals Pra<Xb a X b

Definition 1: Cumulative distribution
The cumulative distribution function of a random variable X is a function F X F X such that such that
F X b=PrXb=PrωΩ| Xωb F X b X b X ω b ω Ω
(1)
Definition 2: Continuous Random Variable
A random variable XX is continuous if
F X b=bf X xdx F X b x b f X x
(2)
and f X x f X x is the probability density function (pdr) (e.g., F X x F X x is differentiable and f X x=ddx F X x f X x x F X x )
Definition 3: Discrete Random Variable
A random variable XX is discrete if it only takes at most countably many points (i.e., F X F X is piecewise constant). The probability mass function (pmf) is defined as
p X x k =PrX= x k = F X x k limit   x (x x k )(x< x k ) F X x p X x k X x k F X x k x x x k x x k F X x
(3)

Two random variables defined on an experiment have joint distribution

F X Y ab=PrXaYb=PrωΩ| (Xωa)(Yωb) F X Y a b X a Y b X ω a Y ω b ω Ω
(4)

Joint pdf can be obtained if they are jointly continuous.

F X Y ab=baf X Y xyd x d y F X Y a b y b x a f X Y x y
(5)
(e.g., f X Y xy=2 F X Y xy x y 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 =PrX= x k Y= y l p X Y x k y l X x k Y y l
(6)

Conditional density function

f Y | X y | x =f X Y xyf X x f Y | X y | x f X Y x y f X x
(7)
for all xx and with f X x>0 f X x 0

## Content actions

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