Lecture9_Spring11

Lecture9_Spring11 - Elec 210: Lecture 9 Important discrete...

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Elec210 Lecture 9 1 Elec 210: Lecture 9 Important discrete random variables Summary of variables you know: Bernoulli • Binomial • Geometric • Discrete Uniform New random variable: Poisson MATLAB commands for plotting probability mass functions and generating discrete random variables
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Elec210 Lecture 9 2 Bernoulli Random Variable The Bernoulli Random Variable is simply a random variable that assumes either value 0 or 1 with probabilities (1- p ) and p . Mean and Variance Used to model Single coin toss Occurrence of an event of interest x (0) 1 (1) XX pp p p =− = [] VAR[ ] (1 ) E Xp X = 1 -p 1 p 0 p X ( x ) pmf 0 0.5 1 0 0.1 0.2 VAR( ) X p
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Suppose a random experiment is repeated n independent times. For each trial, an event A occurs with probability p X, number of times that an event A occurs , is a binomial RV with Mean and Variance Applications Multiple coin flips Occurrence of a property in individuals of a population (e.g. bit errors in a transmission, defective parts in a batch) Elec210 Lecture 9 3 ( ) (1 ) for 0,1,. ... kn k X n pk p p k n k  =− =   Binomial Random Variable [] VAR[ ] ) E Xn p X np p = note that these are just n times the values for the Bernoulli (more on this later)
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X = 3 Binomial Random Variable: Example PMFs Elec210 Lecture 9 4 24 0.2 n p = = 24 0.5 n p = = Symmetric for p=0.5? p = 0.8? = ?
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Elec210 Lecture 9 5 Suppose a random experiment is repeated until an event A occurs. In each repeat, A occurs independently and with probability p . The number of trials until the first success, M , is a geometric RV Mean and variance: Applications Number of customers awaiting service in a queueing system Number of white dots between successive black dots in a scan of a document Number of transmissions required until an error free transmission 1 ( ) (1 ) for 1,2,. ..., k M pk p p k =− = Geometric Random Variable 2 11 [ ] VAR[ ] p EM M p p ==
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Geometric RV: Example PMFs Elec210 Lecture 9 6 0.5 p = 0.3 p = X = 4 = ?
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Elec210 Lecture 9 7 Let M be the number of trials until the first success in a sequence of Bernoulli trials. M is a geometric random variable. Q: What is the conditional probability that it will take k more trials until the first success, given that we have already performed m trials with no success?
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Lecture9_Spring11 - Elec 210: Lecture 9 Important discrete...

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