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Lecture 12 Probability and Random Variables (II)

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

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Lecture 12 Probability and Random Variables (II)

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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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