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Probability & Random Variables: Transformation, Averages, and Moments

Learn how to create new random variables via transformations, compute statistical averages, variance, covariance, correlation, and understand moment generating functions. Explore key concepts in probability theory such as the Rayleigh pdf and conditional expectations.

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Probability & Random Variables: Transformation, Averages, and Moments

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  1. Lecture 12Probability and Random Variables (II) Fall 2008 NCTU EE Tzu-Hsien Sang 1 1

  2. Transformation of R.V. • Create a new r.v. via doing transformation on old r.v. • Example: Sometimes a quantity that we are interested in knowing depends on another quantity which is random…

  3. Example 4.15: X and Y are independent and Gaussian, zero mean and variance=s2. Transform (X, Y) to (R, Q). The Rayleigh pdf

  4. Statistical Averages • Mean (Weighted Average) • The r-th moment

  5. The r-th central moment Variance: • The r-th joint moment • Correlation Note: Independent: FXY(x,y) = FX(x)FY(y) Uncorrelated: E((X-E(X))(Y-E(Y))) = 0 Orthogonal: E(XY) = 0

  6. The r-th joint central moment • Covariance • Correlation coefficient • Conditional expectation • Expectation of functions of r.v.

  7. Moment generating functions • Characteristic functions

  8. Error Function and Q Function

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