Lecture 12 Probability and Random Variables (II)

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Lecture 12Probability and Random Variables (II)

• Fall 2008
• NCTU EE
• Tzu-Hsien Sang

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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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• Example 4.15 X and Y are independent and
  Gaussian, zero mean and variances2. Transform
  (X, Y) to (R, Q).

The Rayleigh pdf
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Statistical Averages

• Mean (Weighted Average)
• The r-th moment

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• 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

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

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• Moment generating functions
• Characteristic functions

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Error Function and Q Function
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