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Unformatted text preview: Sections 2.11 and 5.8 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 25 Gesell data Let X be the age in in months a child speaks his/her first word and let Y be the Gesell adaptive score, a measure of a child’s aptitude (observed later on). Are X and Y related? How does the child’s aptitude change with how long it takes them to speak? Here’s the Gesell score y i and age at first word in months x i data, i = 1 , . . . , 21. x i y i x i y i x i y i x i y i x i y i 15 95 26 71 10 83 9 91 15 102 20 87 18 93 11 100 8 104 20 94 7 113 9 96 10 83 11 84 11 102 10 100 12 105 42 57 17 121 11 86 10 100 In R, we compute r = . 640, a moderately strong negative relationship between age at first word spoken and Gesell score. > age=c(15,26,10,9,15,20,18,11,8,20,7,9,10,11,11,10,12,42,17,11,10) > Gesell=c(95,71,83,91,102,87,93,100,104,94,113,96,83,84,102,100,105,57,121,86,100) > plot(age,Gesell) > cor(age,Gesell) [1] 0.64029 2 / 25 Scatterplot of ( x 1 , y 1 ) , . . . , ( x 21 , y 21 ) 10 15 20 25 30 35 40 60 70 80 90 100 110 120 age Gesell 3 / 25 Random vectors A random vector X = X 1 X 2 . . . X k is made up of, say, k random variables. A random vector has a joint distribution, e.g. a density f ( x ), that gives probabilities P ( X ∈ A ) = integraldisplay A f ( x )d x . Just as a random variable X has a mean E ( X ) and variance var( X ), a random vector also has a mean vector E ( X ) and a covariance matrix cov( X ). 4 / 25 Mean vector & covariance matrix Let X = ( X 1 , . . . , X k ) be a random vector with density f ( x 1 , . . . , x k ). The mean of X is the vector of marginal means E ( X ) = E X 1 X 2 . . . X k = E ( X 1 ) E ( X 2 ) . . . E ( X k ) . The covariance matrix of X is given by cov ( X ) = cov ( X 1 , X 1 ) cov ( X 1 , X 2 ) ··· cov ( X 1 , X k ) cov ( X 2 , X 1 ) cov ( X 2 , X 2 ) ··· cov ( X 2 , X k ) . . . . . . . . . . . . cov ( X k , X 1 ) cov ( X k , X 2 ) ··· cov ( X k , X k ) . 5 / 25 Multivariate normal distribution The normal distribution generalizes to multiple dimensions. We’ll first look at two jointly distributed normal random variables, then discuss three or more. The bivariate normal density for ( X 1 , X 2 ) is given by f ( x 1 , x 2 ) = 1 2 πσ 1 σ 2 radicalbig 1 ρ 2 exp braceleftBigg 1 2(1 ρ 2 ) bracketleftBigg parenleftbigg x 1 μ 1 σ 1 parenrightbigg 2 2 ρ parenleftbigg x 1 μ 1 σ 1 parenrightbiggparenleftbigg x 2 μ 2 σ 2 parenrightbigg + parenleftbigg x 2 μ 2 σ 2 parenrightbigg 2 bracketrightBiggbracerightBigg ....
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 Fall '11
 Staff
 Statistics, Normal Distribution, X1, Multivariate normal distribution

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