Lect2 - Stochastic Process 9/22/2006 Lecture 2 Jointly...

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Unformatted text preview: Stochastic Process 9/22/2006 Lecture 2 Jointly Gaussian Random Variables NCTUEE Summary This lecture reviews several important concepts Gaussian random variables. Specifically, I will discuss: • Gaussian Random Variable • Moment Generating Function • Central Limit Theorem • Jointly Gaussian Random Variables • Joint Gaussian Density Function Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively 2-1 Motivation: Why a special attention to Gaussian RVs? • They are analytically tractable → Preserved by linear systems • Central limit theorem → Gaussian can approximate a large variety of distributions in large samples • Useful as models of communication links 1 Gaussian Random Variables Definition 1 The probability density function (pdf) f X ( x ) for a Gaussian random variable X with mean μ and variance σ 2 , denoted by X ∼ N ( μ,σ 2 ) , is given by f X ( x ) = 1 √ 2 πσ 2 e- ( x- μ ) 2 2 σ 2 (1) for-∞ < x < ∞ . Remarks: • Verify that X has mean μ and variance σ 2 • If X ∼ N ( μ,σ 2 ), then the random variable Z = X- μ σ has a N (0 , 1) distribution, also known as the standard normal . • In general, the random variable Z 1 = aX + b for any real scalars a and b is also Gaussian with mean aμ + b and variance a 2 σ 2 . • A Gaussian random variable can be characterized by its first 2 moments E [ X ] and E [ X 2 ] 2-2 2 Moment Generating Function Definition 2 The moment generating function (MGF) of a continuous ran- dom variable X with pdf f X ( x ) is defined by θ ( t ) , E [ e tX ] = Z ∞-∞ e tX f X ( x ) dx, where t is a complex variable. For a discrete random variable X with proba- bility mass function (pmf) p X ( x ) = P X [ X = x ] , the MGF is defined by θ ( t ) , X i e tx i P X [ X = x i ] . Remarks: (1) It’s similar to the Laplace transform. Thus, in general, there is a one- to-one correspondence between θ ( t ) and f X ( x ). (2) Used for computing moments The n th moment of X can be obtained by E [ X n ] = d n dt n ( θ ( t )) fl fl fl t =0 ....
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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Lect2 - Stochastic Process 9/22/2006 Lecture 2 Jointly...

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