Chapter 5 - Chapter 5 Joint Probability Distributions and...

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Unformatted text preview: Chapter 5 Joint Probability Distributions and Random Samples 5.1 Jointly Distributed Random Variables Def) X,Y : two discrete rv’s Joint pmf p(x,y) : p(x,y)=P(X=x, Y=y) ( , ) x y S 2200 ∈ ( , ) ( ( , ) ) ( , ) x y A P X Y A p x y ∈ ∈ = ∑ Ex 5.1) deductible amount for homeowner’s and automobile Insurances p(x,y) y 0 100 200 x 10 20 .20 .10 .20 .05 .15 .30 p(100,0)=0.2, p(250,100)=0.15, ( 100) (100, 100) (100, 200) ( 250, 100) ( 250, 200) P Y p p p p ≥ = + + + Def) marginal pmf of X and Y: ( ) , ( ) X Y p x p y ( ) ( , ) ( ) ( , ) X y Y x p x p x y p y p x y = = ∑ ∑ Ex 5.2) – Ex 5.1 (100) (100, 0) (100, 100) (100, 200) 0. 5 X p p p p = + + = 1 ( 250) ( 250, 0) ( 250, 100) ( 250, 200) 0. 5 X p p p p = + + = 0. 5 100, 250 ( ) X x p x ot her w i se = = = = = . 25 0, 100 ( ) . 50 200 Y y p y y ow ( 100) (100) (200) .75 Y Y P Y p p ≥ = + = Def) joint pdf of X and Y : f(x,y) ( ( , ) ) ( , ) A P X Y A f x y dxdy ∈ = ∫∫ ( , ) ( , ) b d a c P a X b c Y d f x y dydx ≤ ≤ ≤ ≤ = ∫ ∫ For legitimate pdf, ( , ) 0, ( , ) 1 f x y f x y dxdy ∞-∞ ≥ = ∫ 2 Ex 5.3 ) X – proportion of drive-up is used Y – proportion of walk-up is used 2 6 ( , ) ( ) , , 1 5 f x y x y x y = + ≤ ≤ Legitimate pdf? 1 / 4 1 / 4 2 1 1 6 ( 0 , 0 ) ( ) 4 4 5 0. 0109 P X Y x y dxdy ≤ ≤ ≤ ≤ = + = ∫ ∫ Def) marginal pdf : ( ) , ( ) X Y f x f y ( ) ( , ) ( ) ( , ) X Y f x f x y dy f y f x y dx ∞-∞ ∞-∞ = = ∫ ∫ Ex 5.4 ) - Ex 5.3 1 2 6 ( ) ( ) 5 6 2 , 0 1 5 5 X f x x y dy x x = + = + ≤ ≤ ∫ Ex 5.5 ) A case where possible values of X and Y are dependentEx 5....
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This note was uploaded on 01/06/2011 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue University.

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Chapter 5 - Chapter 5 Joint Probability Distributions and...

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