5.04 - Chapter5,Section4 MultivariateDistributions...

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1 Chapter 5, Section 4   Multivariate Distributions Independence John J Currano, 03/29/2010
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2 Recall that two events A and B are independent if, and only if, P ( A B ) = P ( A ) P ( B ). What should it mean for random variables Y 1 and Y 2 to be independent? Informally : any event A expressible in terms of Y 1 alone should be independent of any event B expressible in terms of Y 2 alone. Thus, at a minimum we should have the following: If y 1 and y 2 are any two real numbers, and if A = event { Y 1 y 1 } and B = event { Y 2 y 2 }, P ( A B ) = P ( A ) P ( B ) ( ) P ( Y 1 y 1 , Y 2 y 2 ) = P ( Y 1 y 1 ) P ( Y 2 y 2 ) or, if Y 1 and Y 2 are jointly distributed with distribution function F ( y 1 , y 2 ), ( ) F ( y 1 , y 2 ) = F 1 ( y 1 ) F 2 ( y 2 ) where F 1 and F 2 are the marginal cdfs of Y 1 and Y 2 , respectively. In fact, this is how we define independence of random variables.
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3 Definition. If Y 1 and Y 2 are jointly distributed random variables with joint cdf F ( y 1 , y 2 ) and marginal cdfs F 1 ( y 1 ) and F 2 ( y 2 ), respectively, then we say that Y 1 and Y 2 are independent if F ( y 1 , y 2 ) = F 1 ( y 1 ) F 2 ( y 2 ) for all real numbers y 1 and y 2 . If Y 1 and Y 2 are not independent, we say that they are dependent . Note. A similar definition is made in the multivariate case: Y 1 , Y 2 , . . . , Y n are independent if their joint cdf factors as the product of the marginal cdfs.
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4 Remark. If: Y 1 and Y 2 are discrete random variables     y 1 and y 2 any real numbers     A = event { Y 1 = y
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5.04 - Chapter5,Section4 MultivariateDistributions...

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