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Random signals and Processes ref: F. G. Stremler, Introduction to Communication Systems 3/e NA • Probability P( A) lim N N • All possible outcomes (A1 to AN) are included N P( A ) 1 i 1 i N AB • Joint probability P( AB) lim N N P( AB) P( B | A) P( A) P( A | B ) P( B ) • Conditional probability N AB N AB N P( AB) P( B | A) NA NA / N P( A) Ya Bao N AB N AB N P( AB) P( A | B ) NB NB / N P( B ) Fundamentals of Communications theory 1 Examples • Bayes’ theorem P( B ) P( A | B ) P( B | A) P( A) • Random 2/52 playing cards. After looking at the first card, P(2nd is heart)=? if 1st 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) Ya Bao Fundamentals of Communications theory 2 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 nx a F ( a ) P( x a ) lim ( ) n n • Properties of F(a) • Nondecreasing, • 0<=F(a)<=1, F ( ) 0 F ( ) 1 Ya Bao Fundamentals of Communications theory 3 Probability density function (PDF) dF ( a ) f ( x) |a x da Properties of PDF f ( x ) 0. f ( x)dx F () 1 Ya Bao Fundamentals of Communications theory 4 Discrete and continuous distributions • Discrete: random variable has M discrete values CDF or F(a) was discontinuous as a increase Digital communications M PDF f ( x ) P( x ) ( x x ) i 1 i i M is the number of discretely events L CDF F ( a ) P( xi ) i 1 L is the largest integer such that xL a, L M Ya Bao Fundamentals of Communications theory 5 • 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 M y [h( x )] h( xi ) P( xi ) i 1 Normalized average power P= y2i p(yi) i Ya Bao Fundamentals of Communications theory 6 Important distributions • • • • • Binomial Poisson Uniform Gaussian Sinusoidal Ya Bao Fundamentals of Communications theory 7 Ya Bao Fundamentals of Communications theory 8