stat400lec18 - Statistics 400 Distributions of Two Random...

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Statistics 400 Distributions of Two Random Variables We often have observations of two random variables. Example: Weight and Height Intelligence and diligence Technical merits and presentation Discrete random varaibles Joint probability mass function of (X,Y) x 1 x 2 x 3 x 4 …… x k y 1 f(x 1, y 1 ) f(x 2, y 1 ) f(x 3, y 1 ) f(x 4, y 1 ) …… f(x k, y 1 ) y 2 f(x 1, y 2 ) f(x 2, y 2 ) f(x 3, y 2 ) f(x 4, y 2 ) …… f(x k, y 2 ) y 3 f(x 1, y 3 ) f(x 2, y 3 ) f(x 3, y 3 ) f(x 4, y 3 ) …… f(x k, y 3 ) …. …. …. …. …… …. y m f(x 1, y m ) f(x 2, y m ) f(x 3, y m ) f(x 4, y m ) …… f(x k, y m ) Properties of joint p.m.f f(x,y) : (a) 1 f(x,y) 0; ( b ) ∑∑ = 1 ) , ( y x f ; ( c ) P(X=x,Y=y)=f(x,y). Example: f(x,y)=(x+y)/21 for x=1,2,3 and y=1,2. Is this a valid joint pmf? Ping Ma Lecture 18 Fall 2005 - 1 -

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Marginal probability mass function Given the joint probability mass function of (X,Y), how to find the probability mass function of X alone or Y alone? = = = y x X P y x f x f ) ( ) , ( ) ( 1 ; = = = x y Y P y x f y f ) ( ) , ( ) ( 2 .
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