CPE 332 Computer Engineering Mathematics II Part II

CPE 332 Computer Engineering Mathematics II Part II, Chapter 4 Probability and Random Variable (Review)

Today Topics • HW 3 Due • • • Probability Random Variable PDF Expectation Concept of Random Process

Example: Dice Roll • Sample Space = {1, 2, 3, 4, 5, 6} • P(1)=P(2)= … =P(6)=1/6 – Laplace Definition of Probability – ���� Member �� Sample Space ���������� • A = Even • A = {2, 4, 6} • P(A) = |{2, 4, 6}|/|S| = 3/6 = ½

Mutually Exclusive A A B B • • ME A B = A B=A+B P(A B)=P(A)+P(B) • Non ME • A B=A+B- A B – Inclusion-Exclusion Principle

3 AXIOMS OF Probability • 1. P(S) = 1 – ��������������� Sample Space • 2. 0 =< P(A) =< 1 – ������ Probability ������� 0 ��� 1 • 3. ME: P(A+B)=P(A)+P(B) – Mutually Exclusive; Probability ��� Union ��� Event �������� Probability ���� Event

Bayes Rule E 1 E 3 E 6 A E 4 E 2 E 5 E 7 E 8 E 9

Properties

Random Variables • ������������ (Sample Space)��� Infinite Set �������������� – ������ Continuous Random Variable • �������� Finite Set ��������� Integer – ������ Discrete Random Variable • Probability ����������� ��� Probability ��� Random Variable ����������� • ������������� RV ������ Plot ��� Probability(y-axis) ����� Random Variable(xaxis) – Cumulative Distribution Function (CDF) – Probability Density Function (PDF)

CDF: Cumulative Distribution Function of RV X FX(x) 1. 0 FX (10) = P[X ≤ 10] x = 10 CDF of Normal (Gaussian) Distribution: Continuous x

CDF Properties

PDF: Probability Density Function f(x) Area = ∫f(x)dx = 1 x PDF of Normal(Gaussian) Distribution : Continuous

Properties of PDF

f(x) PMF x F(x) CDF x

Statistical Average

Importance Expectation

Joint Moment

Correlation • G(x, y) = XY

Covarience

Correlation and Covarience

PDF ทสำคญ 1. 0 P[X=x] P[X<=x] p 0 q 1 x

Binomial Distribution b(k; 10, 0. 5) b(k; 10, 0. 2)

Geometric Distribution P=0. 5 P=0. 2

Uniform Distribution f(x) 1/(b-a) a b x

Gaussian Distribution

Gaussian Area = 1

Jointly Gaussian: X, Y

P=0 Volumn = 1

P = 0. 5

P = 0. 9

P = 0. 95

P = -0. 99

Exponential Distribution

Exponential Distribution A=1 Area = 1

Poisson Distribution

Lambda = 1

Lambda = 3

Lambda = 5

Lambda = 8

Lambda = 12

Lambda = 18

Homework IV Due Next Week • Download ��� Web • Next Week Chapter V – Random Process – Mar. Kov Process and Mar. Kov Chain