EE 505 Multiple Random Variables (cont)

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EE 505 Multiple Random Variables (cont) copyright Robert J. Marks II

Functions of Several Random Variables Types of Transformations A Single Function of n RV’s Functions of n RV’s copyright Robert J. Marks II

Functions of Several Random Variables Functions of n RV’s Functions of Several Random Variables Assume Invertability copyright Robert J. Marks II

Functions of Several Random Variables (cont) Transformation Jacobian Functions of Several Random Variables (cont) copyright Robert J. Marks II

Functions of Several Random Variables (cont) Note: copyright Robert J. Marks II

Functions of Several Random Variables (cont) Thus: One dimensional transformation is a special case. copyright Robert J. Marks II

Functions of Several Random Variables (Gaussian Example) X and Y are two i.i.d. Zero mean Gaussian RV’s: x y v w Inversions: copyright Robert J. Marks II

Functions of Several Random Variables (Gaussian Example) Jacobian Then: copyright Robert J. Marks II

Functions of Several Random Variables (Gaussian Example) Jacobian Where Thus copyright Robert J. Marks II

Functions of Several Random Variables (Gaussian Example) Marginals: Rayleigh RV Uniform RV Relation to velocities in an ideal gas. copyright Robert J. Marks II

Functions of Several Random Variables (Auxiliary Function Example) Auxiliary Functions: Finding V = g(X,Y). Define auxiliary function W. Evaluate joint fVW(v,w). Generate marginal Example: Student’s t RV (p.230) copyright Robert J. Marks II

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