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Unformatted text preview: Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Multiple Random Variables Michael Akritas Michael Akritas Multiple Random Variables Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Joint Distribution Functions Independent Random Variables Moment Generating Functions Conditional Distributions Hierarchical Models Covariance and Correlation: Quantifying Dependence Michael Akritas Multiple Random Variables Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Independent Random Variables Moment Generating Functions Conditional Distributions Hierarchical Models I The joint cumulative distribution function of X and Y is F ( a , b ) = P [ X ≤ a , Y ≤ b ] ,∞ < a , b < ∞ . Not as handy to use as in the univariate case. For example: P ( a 1 < X ≤ b 1 , a 2 < Y ≤ b 2 ) = F ( b 1 , b 2 ) F ( a 1 , b 2 ) F ( b 1 , a 2 ) + F ( a 1 , a 1 ) while P ( ( X , Y ) ∈ { ( x , y ) : x 2 + y 2 ≤ r 2 } ) is not expressible in terms of F ( x , y ). Michael Akritas Multiple Random Variables Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Independent Random Variables Moment Generating Functions Conditional Distributions Hierarchical Models If X and Y are jointly continuous such probabilities can be given in terms of the joint probability density function, which is a function f : R 2 → R such that P (( X , Y ) ∈ A ) = Z A Z f ( x , y ) dxdy , holds ∀ A ⊆ R 2 In particular, I F ( x , y ) = R ∞∞ R ∞∞ f ( s , t ) dsdt , and by the F.T.C. I ∂ 2 F ( x , y ) ∂ x ∂ y = f ( x , y ), at continuity points of f . Michael Akritas Multiple Random Variables Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Independent Random Variables Moment Generating Functions Conditional Distributions Hierarchical Models If X and Y are jointly discrete probabilities P (( X , Y ) ∈ A ) can be given in terms of the joint probability mass function, which is defined as p ( x , y ) = P ( X = x , Y = y ) The marginal distributions of X and Y are given by: 1. F X ( a ) = F ( a , ∞ ), F Y ( b ) = F ( ∞ , b ). 2. p X ( x ) = ∑ y : p ( x , y ) > p ( x , y ), p Y ( y ) = ∑ x : p ( x , y ) > p ( x , y ). 3. f X ( x ) = R ∞∞ f ( x , y ) dy , f Y ( y ) = R ∞∞ f ( x , y ) dx . Michael Akritas Multiple Random Variables Outline Joint Distribution Functions Covariance and Correlation: Quantifying Dependence Independent Random Variables Moment Generating Functions Conditional Distributions Hierarchical Models Example 45% of customers will purchase a basic flat screen TV, 15% will purchase a high resolution flat screen, and the rest will either just browse or buy something else. Out of the next 5 customers, let X 1 denote the number who buy the basic model, and X 2 the number who buy the high resolution model....
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This note was uploaded on 01/23/2011 for the course STAT 418 at Penn State.
 '08
 G.JOGESHBABU
 Correlation, Covariance, Probability, Variance

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