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Review of Probability Theory Review Session 1 EE384X Winter 2006 EE384x 1 Random Variable A random experiment with set of outcomes Random variable is a function from set of outcomes to real numbers Winter 2006 EE384x 2 Example Indicator random variable: A : A subset of Winter 2006 is called an event EE384x 3 CDF and PDF Discrete random variable: The possible values are discrete (countable) Continuous random variable: The rv can take a range of values in R Cumulative Distribution Function (CDF): PDF and PMF: Winter 2006 EE384x 4 Expectation and higher moments Expectation (mean): if X>0 : Variance: Winter 2006 EE384x 5 Two or more random variables Joint CDF: Covariance: Winter 2006 EE384x 6 Independence For two events A and B: Two random variables IID : Independent and Identically Distributed Winter 2006 EE384x 7 Useful Distributions Winter 2006 EE384x 8 Bernoulli Distribution The same as indicator rv: IID Bernoulli rvs (e.g. sequence of coin flips) Winter 2006 EE384x 9 Binomial Distribution Repeated Trials: Number of times an event A happens among n trials has Binomial distribution Repeat the same random experiment n times. (Experiments are independent of each other) (e.g., number of heads in n coin tosses, number of arrivals in n time slots,…) Binomial is sum of n IID Bernouli rvs Winter 2006 EE384x 10 Mean of Binomial Note that: Winter 2006 EE384x 11 Binomial - Example 0.45 n=4 0.4 0.35 p=0.2 n=10 0.3 0.25 n=20 0.2 n=40 0.15 0.1 0.05 0 Winter 2006 1 2 3 4 5 6 EE384x 7 8 9 10 11 12 12 Binomial – Example (ball-bin) There are B bins, n balls are randomly dropped into bins. : Probability that a ball goes to bin i : Number of balls in bin i after n drops Winter 2006 EE384x 13 Multinomial Distribution Generalization of Binomial Repeated Trails (we are interested in more than just one event A) A partition of W into A1,A2,…,Al Xi shows the number of times among n trails. Winter 2006 EE384x Ai occurs 14 Poisson Distribution Used to model number of arrivals Winter 2006 EE384x 15 Poisson Graphs 0.5 l=.5 0.45 0.4 l=1 0.35 0.3 0.25 l=4 0.2 l=10 0.15 0.1 0.05 0 Winter 2006 0 5 10 EE384x 15 16 Poisson as limit of Binomial Poisson is the limit of Binomial(n,p) as Let Winter 2006 EE384x 17 Poisson and Binomial 0.4 n=5,p=4/5 0.35 Poisson(4) 0.3 0.25 n=10,p=.4 0.2 n=20, p=.2 0.15 0.1 n=50,p=.08 0.05 0 Winter 2006 0 1 2 3 4 5 EE384x 6 7 8 9 10 18 Geometric Distribution Repeated Trials: Number of trials till some event occurs Winter 2006 EE384x 19 Exponential Distribution Continuous random variable Models lifetime, inter-arrivals,… Winter 2006 EE384x 20 Minimum of Independent Exponential rvs Winter 2006 : Independent Exponentials EE384x 21 Memoryless property True for Geometric and Exponential Dist.: The coin does not remember that it came up tails l times Root cause of Markov Property. Winter 2006 EE384x 22 Proof for Geometric Winter 2006 EE384x 23 Characteristic Function Moment Generating Function (MGF) For continuous rvs (similar to Laplace xform) For Discrete rvs (similar to Z-transform): Winter 2006 EE384x 24 Characteristic Function Can be used to compute mean or higher moments: If X and Y are independent and T=X+Y Winter 2006 EE384x 25 Useful CFs Bernoulli(p) : Binomial(n,p) : Multinomial: Poisson: Winter 2006 EE384x 26