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BU EECE 522 - Review of Probability

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Review of Probability 1 28 Random Variable Definition Numerical characterization of outcome of a random event Examples 1 Number on rolled dice 2 Temperature at specified time of day 3 Stock Market at close 4 Height of wheel going over a rocky road 2 28 Random Variable Non examples 1 Heads or Tails on coin 2 Red or Black ball from urn But we can make these into RV s Basic Idea don t know how to completely determine what value will occur Can only specify probabilities of RV values occurring 3 28 Two Types of Random Variables Random Variable Discrete RV Die Stocks Continuous RV Temperature Wheel height 4 28 PDF for Continuous RV Given Continuous RV X What is the probability that X x0 Oddity P X x0 0 Otherwise the Prob Sums to infinity Need to think of Prob Density Function PDF pX x xo The Probability density function of RV X xo x P x0 X x0 area shown xo x o p X x dx 5 28 Most Commonly Used PDF Gaussian A RV X with the following PDF is called a Gaussian RV p X x 1 2 e x m 2 2 2 m are parameters of the Gaussian pdf m Mean of RV X Standard Deviation of RV 2 Variance of RV X X Note 0 Notation When X has Gaussian PDF we say X N m 2 6 28 Zero Mean Gaussian PDF Generally take the noise to be Zero Mean p x x 1 2 e x 2 2 2 7 28 Effect of Variance on Guassian PDF pX x Area within 1 of mean 0 683 68 3 x m pX x x Small Small Variability Small Uncertainty x pX x Large x Large Variability Large Uncertainty 8 28 Why Is Gaussian Used Central Limit theorem CLT The sum of N independent RVs has a pdf that tends to be Gaussian as N So What Here is what Electronic systems generate internal noise due to random motion of electrons in electronic components The noise is the result of summing the random effects of lots of electrons CLT applies Guassian Noise 9 28 p XY x y Joint PDF of RVs X and Y Describes probabilities of joint events concerning X and Y For example the probability that X lies in interval a b and Y lies in interval a b is given by bd Pr a X b and c Y d p XY x y dxdy ac This graph shows the Joint PDF Graph from B P Lathi s book Modern Digital Analog Communication Systems 10 28 Conditional PDF of Two RVs When you have two RVs often ask What is the PDF of Y if X is constrained to take on a specific value In other words What is the PDF of Y conditioned on the fact X is constrained to take on a specific value Ex Husband s salary X conditioned on wife s salary 100K First find all wives who make EXACTLY 100K how are their husband s salaries distributed Depends on the joint PDF because there are two RVs but it should only depend on the slice of the joint PDF at Y 100K Now we have to adjust this to account for the fact that the joint PDF even its slice reflects how likely it is that X 100K will occur e g if X 105 is unlikely then pXY 105 y will be small so if we divide by pX 105 we adjust for this 11 28 Conditional PDF cont Thus the conditional PDFs are defined as slice and normalize p XY x y pY X y x p X x 0 p X x 0 otherwise p XY x y p X Y x y pY y 0 x is held fixed pY y 0 otherwise y is held fixed slice and normalize y is held fixed This graph shows the Conditional PDF Graph from B P Lathi s book Modern Digital Analog Communication Systems 12 28 Independent RV s Independence should be thought of as saying that neither RV impacts the other statistically thus the values that one will likely take should be irrelevant to the value that the other has taken In other words conditioning doesn t change the PDF pY X x y x p XY x y pY y p X x p XY x y p X Y y x y p X x pY y 13 28 Independent and Dependent Gaussian PDFs y Independent zero mean Independent non zero mean Contours of pXY x y x y x Dependent y x If X Y are independent then the contour ellipses are aligned with either the x or y axis Different slices give same normalized curves Different slices give different normalized curves 14 28 An Independent RV Result RV s X Y are independent if p XY x y p X x pY y Here s why p XY x y p X x pY y pY X x y x pY y p X x p X x 15 28 Characterizing RVs PDF tells everything about an RV but sometimes they are more than we need know So we make due with a few Characteristics Mean of an RV Variance of an RV Correlation of RVs Describes the centroid of PDF Describes the spread of PDF Describes tilt of joint PDF Mean Average Expected Value Symbolically E X 16 28 Motivating Idea of Mean of RV Motivation First w Data Analysis View Consider RV X Score on a test Data x1 x2 xN Possible values of RV X V0 V1 V2 V100 0 1 2 100 Test Average x N x i 1 i N N 0V0 N1V1 N nV100 100 N i Vi N N i 0 Ni of scores of value Vi n N i Total of scores N i 1 This is called Data Analysis View But it motivates the Data Modeling View P X Vi Statistics Probability 17 28 Theoretical View of Mean Data Analysis View leads to Probability Theory For Discrete random Variables Data Modeling n E X xi PX xi n 1 Probability Function This Motivates form for Continuous RV E X x p X x dx Notation E X X Probability Density Function Shorthand Notation 18 28 Aside Probability vs Statistics Statistics Given a set of Data Determine how the data did behave Probability Theory Given a PDF Model Describe how the data will likely behave E X x p X x dx PDF Law of Large Numbers 1 Avg N n xi i 1 Data Dummy Variable There is no DATA here The PDF models how the data will likely behave There is no PDF here The Statistic measures how the data did behave 19 28 Variance of RV There are similar Data vs Theory Views here But let s go right to the theory Variance Characterizes how much you expect the RV to Deviate Around the Mean Variance 2 E X m x 2 x m x 2 p X x dx Note If zero mean 2 E X 2 x 2 p X x dx …


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BU EECE 522 - Review of Probability

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