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Material Type: Notes; Professor: Koushanfar; Class: RANDOM SIGNALS IN ELECTRICAL ENGINEERING SYSTEMS; Subject: Electrical & Comp. Engineering; University: Rice University; Term: Fall 2008;

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Download Derived Distributions, Covariance, Correlation and Convolution | ELEC 303 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! 10/2/2008 1 ELEC 303, Koushanfar, Fall’08 ELEC 303 – Random Signals Lecture 11 – Derived distributions, covariance, correlation and convolution Dr. Farinaz Koushanfar ECE Dept., Rice University Oct 1, 2008 ELEC 303, Koushanfar, Fall’08 Lecture outline • Reading: 4.1-4.2 • Derived distributions • Sum of independent random variables • Covariance and correlations 10/2/2008 2 ELEC 303, Koushanfar, Fall’08 Derived distributions • Consider the function Y=g(X) of a continuous RV X • Given PDF of X, we want to compute the PDF of Y • The method – Calculate CDF FY(y) by the formula – Differentiate to find PDF of Y ELEC 303, Koushanfar, Fall’08 Example 1 • Let X be uniform on [0,1] • Y=sqrt(X) • FY(y) = P(Yy) = P(Xy2) = y2 • fY(y) = dF(y)/dy = d(y2)/dy = 2y 0 y1 10/2/2008 5 ELEC 303, Koushanfar, Fall’08 More on strictly monotonic case ELEC 303, Koushanfar, Fall’08 Example 4 • Two archers shoot at a target • The distance of each shot is ~U[0,1], independent of the other shots • What is the PDF for the distance of the losing shot from the center? 10/2/2008 6 ELEC 303, Koushanfar, Fall’08 Example 5 • Let X and Y be independent RVs that are uniformly distributed on the interval [0,1] • Find the PDF of the RV Z? ELEC 303, Koushanfar, Fall’08 Sum of independent RVs - convolution 10/2/2008 7 ELEC 303, Koushanfar, Fall’08 X+Y: Independent integer valued ELEC 303, Koushanfar, Fall’08 X+Y: Independent continuous 10/2/2008 10 ELEC 303, Koushanfar, Fall’08 Covariance • Covariance of two RVs is defined as follows • An alternate formula: Cov(X,Y) = E[XY] – E[X]E[Y] • Properties – Cov(X,X) = Var(X) – Cov(X,aY+b) = a Cov(X,Y) – Cov(X,Y+Z) = Cov(X,Y) + Cov (Y,Z) ELEC 303, Koushanfar, Fall’08 Covariance and correlation • If X and Y are independent E[XY]=E[X]E[Y] • So, the cov(X,Y)=0 • The converse is not generally true!! • The correlation coefficient of two RVs is defined as • The range of values is between [-1,1] 10/2/2008 11 ELEC 303, Koushanfar, Fall’08 Variance of the sum of RVs • Two RVs: • Multiple RVs