1. Ya Bao Fundamentals of Communications theory1 Random signals and Processes ref: F. G. Stremler, Introduction to Communication Systems 3/e Probability All possible outcomes (A 1 to A N ) are included Joint probability Conditional probability

2. Ya Bao Fundamentals of Communications theory2 Examples Bayes’ theorem Random 2/52 playing cards. After looking at the first card, P(2 nd is heart)=? if 1 st is or isn’t heart Probability of two mutually exclusive events P(A+B)=P(A)+P(B) If the events are not mutually exclusive P(A+B)=P(A)+P(B)-P(AB)

3. Ya Bao Fundamentals of Communications theory3 Random variables A real valued random variable is a real-value function defined on the events of the probability system. Cumulative distribution function (CDF) of x is Properties of F(a) Nondecreasing, 0<=F(a)<=1,

4. Ya Bao Fundamentals of Communications theory4 Probability density function (PDF) Properties of PDF

5. Ya Bao Fundamentals of Communications theory5 Discrete and continuous distributions Discrete: random variable has M discrete values CDF or F(a) was discontinuous as a increase Digital communications PDF CDF

6. Ya Bao Fundamentals of Communications theory6 Continuous distributions: if a random variable is allowed to take on any value in some interval. CDF and PDF would be continuous functions. Analogue communications, noise. Expected value of a discretely distributed random variable Normalized average power

7. Ya Bao Fundamentals of Communications theory7 Important distributions Binomial Poisson Uniform Gaussian Sinusoidal

8. Ya Bao Fundamentals of Communications theory8