notes4 - Stat 430/Math468 Notes #4 Random Vectors and...

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Stat 430/Math468 – Notes #4 Chapter 2 Random Vectors and Matrices A random vector (or matrix) is a vector (or matrix) whose elements are random variables. Suppose 1 ( ,..., )' p X X = X is a random vector with joint PDF/PMF 1 ( ,. .., ) p f xx and each element has marginal PDF/PMF i X () ii f x with marginal mean ( ) , if is a continuous r.v. ( ), if is a discrete r.v. i ii i i i i all x xf x dx x EX xf x x μ == and marginal variance 2 22 2 ( ) ( ) if is a continuous r.v. () [ ] ( ) if is a discrete r.v. i i i i i i i i i i i i all x xf x d x x Var X E X x x σσ = = The covariance between any pair of random variables, , i X j X measures their the linear relationship. It is calculated through their joint PDF/PMF (, ) ij i j f xx . ( ) ( , ), if , are continuous r.v.'s (, ) [ ( ) ( ) ] ( ) ( , ) , if , are di ij j j i j i j i j ij i j i i j j j j i j i j all x all x xxf x x d x d x Cov X X E X X x x μμ σμ −− = ∫∫ ∑∑ screte r.v.'s The correlation coefficient between and i X j X is defined as ij ij ii jj σ ρ = . The p random variables 1 ,..., p X X are said to be (mutually statistically) independent , if their joint PDF/PMF can be written as the product of their marginal PDF/PMF’s, that is 12 1122 ( , ,..., ) ( ) ( )... ( ) pp p fxx x fx f x f x = , for all p-tuples ( , ,..., p x ). Specially, two random variables , i X j X are said to be (mutually statistically) independent , if (, ) () ( ) ij i j i i j j = , for all pairs ( , x x ). Two random variables , i X j X are said to be (mutually statistically) uncorrelated , if , which is equivalent to (, )0 ij i j Cov X X ) ( ) ( i j EXX EX EX = . In general, independent uncorrelated, but uncorrelated independent. / 1
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The (population) mean vector μ , (population) variance-covariance matrix , and (population) correlation matrix of a random vector Σ ρ The mean vector: 11 22 () pp EX E μ ⎛⎞ ⎜⎟
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This note was uploaded on 03/07/2010 for the course STAT 430 taught by Professor Hua during the Spring '10 term at University of Illinois at Urbana–Champaign.

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notes4 - Stat 430/Math468 Notes #4 Random Vectors and...

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