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332:421 Communications Engineering Lecture Notes 9-11-2000 Probability Review Experiment = Procedure + Observation Sample space (S) = mutually exclusive, collectively exhaustive, finest grain set of all possible outcomes. Random Variable (X): is a function that maps an element of the sample space s Є S to R Random Process is a mapping of the sample outcome s Є S to a waveform X(t, s) Example: roll a die. Result of the roll is N. N Є {1,...,6} If N = n, then let X(t,n) = Xn(t) = { n 0 t ≥ 0 o.w. roll a die an infinite number of times. We generate an infinite sequence S = [n1, n2, n3, …] S = [1,3,4,6,1,1,6,…] 7 6 5 4 X(t ) 3 2 1 0 0 1 2 3 4 5 6 7 8 Map S to the waveform X(t, s) such that: X(t) = { 0 t ≥ 0 nk k-1 < t ≤ k, k = 1,2,3,…… So knowing X(t0) may or may not help you predict X() when t1 ≠ t0 if you know the sample outcome, you know everything. Description: If we have a random variable X; with describe it with a cumulative distribution function: FX(x) = P[X ≤ x] Probability Density Function fX(x) = dfX(x)/dx P[x < X ≤ x + dx] = fX(x) dx If an experiment produces multiple random variables S → X1, X2,..,Xn Joint c.d.f FX1(x1xn - xn) = P[X1 ≤ x1, X2 ≤ x2, X3 ≤ x3,…, Xn ≤ xn] Joint p.d.f fX1...xn(x1...xn) = δfX1...xn(x1...xn)/δx1...δxn P[x1 ≤ X1 ≤ y1, x2 ≤ X2 ≤ y2,..., xn ≤ Xn ≤ yn] = ∫...∫ fX1...xn(x1...xn)δx1...δxn X1...Xn = n r.v’s (random vectors) FX(x) = FX1...xn(x1...xn) Random Process X(t) sampled at time instants X = [X(t1), X(t2),..., X(tk)] Random Process Model: given t1...tk, a model tells you the joint CDF or PDF Roll a die an infinite number of times X(t1) = the result of roll 11. equally likely to be 1,2,..,6 Given, t1 = 11.1 and t2 = 12.2 fX(t1)X(t2)(x1,x2) = P[X(t1) ≤ x1, X(t2) ≤ x2] = P[X(t1) ≤ x1] P[X(t2) ≤ x2] = FX(t1)(x1) FX(t2)(x2) Given, t1 = 11.1 and t2 = 11.3 fX(t1)X(t2)(x1,x2) = P[roll 11 ≤ x1, roll 11 ≤ x2] = P[roll 11 ≤ min(x1, x2)] stationarity When is a random process independent of time? X(t) is a stationary random process if given any time instants t1...tk and time offsets T, then fX(t1)...x(tk)(x1...xk) = fX(t1+T)...X(tk+T)(x1...xk) at k = 1, fX(t1)(x1) = fX(t1+T)(x1) = fX(0)(x1) = fX(x1) for all T at k = 2, fX1(t1)X2(t2)(x1,x2) = fX(t1+T)X(t2+T)(x1,x2) substituting: t2 = t1 + τ and T = -t1 therefore: fX(t1)X(t1+τ)(x1,x2) = fX(0)X(τ)(x1,x2) X(t,s) is a random process Sample function So X(t,So) = xo(t) ← some random output waveform Time average: xo(t) = limT → ∞ 1/T∫xo(t)dt if we restart the experiment n times on experiment I Xi = X(33) X = 1/n∑ xi As n → ∞, X → E[X] (expected value) In general x(t) ≠ X → E[X], x(t) is an ergodic process when they are equal.