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Chapter 8 Probability and Random variables

Chapter 8 Probability and Random variables. F. G. Stremler , Introduction to Communication Systems 3/e. Probability All possible outcomes (A 1 to A N ) are included Joint probability Conditional probability. Bayes ’ theorem

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Chapter 8 Probability and Random variables

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  1. Chapter 8 Probability and Random variables F. G. Stremler, Introduction to Communication Systems 3/e Ya Bao

  2. Probability • All possible outcomes (A1 to AN) are included • Joint probability • Conditional probability Ya Bao

  3. Bayes’ theorem • Random 2/52 playing cards. After looking at the first card, P(2nd is heart)=? if 1st is or isn’t heart A: a heart on the 1st; B: a heart on the 2nd; C: no heart on the 1st P(B|A) = 12/51; P(B|C) = 13|51 • 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

  4. 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, Ya Bao

  5. Probability density function (PDF) Properties of PDF Ya Bao

  6. Tutorial Q.2 Consider the experiment that consists in the rolling of two honest dice. The random variable X is assigned to the sum of the numbers showing up to the two dice. Determine and plot the cumulative distribution function (CDF) and the probability distribution function (pdf) of X. Ya Bao

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  8. Discrete and continuous distributions • Discrete: random variable has M discrete values CDF or F(a) was discontinuous as a increase. Digital communications PDF CDF Ya Bao

  9. 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 Ya Bao

  10. example The normalized average power, P, is given by: i.e. (0.5)2 0.2 + (0.4)2 0.3 + (0.2)2 0.1 + (0.1)2 0.4 = 0.106/100 W = 1.06 mW A discrete random signal, y(t), can take one of the four predefined voltage levels, y1 = 0.5 V, y2 = 0.4 V, y3 = 0.2 V, and y4 = 0.1 V. Assume that these levels occur with probabilities, p(y1) = 0.2, p(y2) = 0.3, p(y3) = 0.1, and p(y4) = 0.4. Calculate the average power delivered by y(t) into a 100Ω resistive load. Ya Bao

  11. Important distributions • Binomial : discrete • Poisson: discrete Ya Bao

  12. Uniform: continuous, a random variable that is equally likely to take on any value within a given range. Gaussian (normal): continuous Normalised Gaussian pdf having zero mean and unit variance Sinusoidal: continuous Ya Bao

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