lecture8_Stat704

# lecture8_Stat704 - Sections 2.11 and 5.8 Timothy Hanson...

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 12/14/2011 for the course STAT 704 taught by Professor Staff during the Fall '11 term at South Carolina.

### Page1 / 25

lecture8_Stat704 - Sections 2.11 and 5.8 Timothy Hanson...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online