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Review of Probability Theory

Module by: Behnaam Aazhang

Summary: (Blank Abstract)

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 information and the transmission media. Interestingly, modeling both digital or analog information and many physical media requires a probabilistic setting. In this chapter and in the next one we will review the theory of probability, model random signals, and characterize their behavior as they traverse through deterministic systems disturbed by noise and interference. In order to develop practical models for random phenomena we start with carrying out a random experiment. We then introduce definitions, rules, and axioms for modeling within the context of the experiment. 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. Such outcomes could be an abstract description in words. A scientific experiment should indeed be repeatable where each outcome could naturally have an associated probability of occurrence. This is defined formally as the ratio of the number of times the outcome occurs 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.
Figure2-1.png
Figure 1
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 ω i = ii dots on the face of the dice.
X ω i =i X ω i i

Distributions

Probability assignments on intervals a<X≤b a X b
Definition 1: Cumulative distribution
The cumulative distribution function of a random variable XX is a function FXℝ↦ℝ F X ↦ such that
FXb=PrX≤b=Pr{ω∈Ω|Xω≤b} F X b X b ω Ω X ω b (1)
Figure2-2.png
Figure 2
Definition 2: Continuous Random Variable
A random variable XX is continuous if the cumulative distribution function can be written in an integral form, or
FXb=∫-∞bfXxdx F X b x b f X x (2)
and fXx f X x is the probability density function (pdf) (e.g., FXx F X x is differentiable and fXx=ddxFXx 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., FX· F X · is piecewise constant). The probability mass function (pmf) is defined as
pX x k =PrX= x k =FX x k -limx→ x k ∧x< x k FXx 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
FXYab=PrX≤aY≤b=Pr{ω∈Ω|Xω≤a∧Yω≤b} F X Y a b X a Y b ω Ω X ω a Y ω b (4)
Figure2-4.png
Figure 3
Joint pdf can be obtained if they are jointly continuous
FXYab=∫-∞b∫-∞afXYxydxdy F X Y a b y b x a f X Y x y (5)
(e.g., fXYxy=∂2∂x∂yFXYxy f X Y x y x y F X Y x y )
Joint pmf if they are jointly discrete
pXY 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 =fXYxyfXx f Y | X y | x f X Y x y f X x (7)
for all xx with fXx>0 f X x 0 otherwise conditional density is not defined for those values of xx with fXx=0 f X x 0
Two random variables are independent if
fXYxy=fXxfYy f X Y x y f X x f Y y (8)
for all x∈ℝ x and y∈ℝ y . For discrete random variables,
pXY x k y l =pX x k pY y l p X Y x k y l p X x k p Y y l (9)
for all kk and ll.

Moments

Statistical quantities to represent some of the characteristics of a random variable.
gX¯=EgX=∫-∞∞gxfXxdxifcontinuous∑kg x k pX x k ifdiscrete g X g X x g x f X x continuous k k g x k p X x k discrete (10)
  • Mean
    μ X =X¯ μ X X (11)
  • Second moment
    EX2=X2¯ X 2 X 2 (12)
  • Variance
    VarX=σX2=X- μ X 2¯=X2¯- μ X 2 Var X X X μ X 2 X 2 μ X 2 (13)
  • Characteristic function
    Φ X u=ⅇⅈuX¯ Φ X u u X (14)
    for u∈ℝ u , where ⅈ=-1 1
  • Correlation between two random variables
    R X Y =X Y * ¯=∫-∞∞∫-∞∞x y * fXYxydxdyif X and Y are jointly continuous∑k∑l x k y l * pXY x k y l ifX and Y are jointly discrete R X Y X Y * y 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 (15)
  • Covariance
    C X Y =CovXY=X- μ X Y- μ Y *¯= R X Y - μ X μ Y * C X Y Cov X Y X μ X Y μ Y * R X Y μ X μ Y * (16)
  • Correlation coefficient
    ρ X Y =CovXY σ X σ Y ρ X Y Cov X Y σ X σ Y (17)
Definition 4: Uncorrelated random variables
Two random variables XX and YY are uncorrelated if ρ X Y =0 ρ X Y 0 .

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