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Lecture_9

# Lecture_9 - 1 INDEPENDENT RANDOM VARIABLES Lecture 9...

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1 INDEPENDENT RANDOM VARIABLES Lecture 9 ORIE3500/5500 Summer2009 Chen Class Today Independent Random Variables Conditional Distribution 1 Independent Random Variables As the name suggests, two random variables are independent if they are not related with each other in any way. Having informaion on one of them does not reveal anything about the other. It is deﬁned as follows: X 1 ,...,X n are said to be independent if for any sets A 1 ,...,A n , P [ X 1 A 1 ,...,X n A n ] = P [ X 1 A n ] ··· P [ X n A n ] . Recall the deﬁnition of independence of events. Here, for every set A 1 ,...,A n , B 1 = [ X 1 A 1 ] ,...,B n = [ X n A n ] are events. For any r sets B k 1 ,...,B k r chosen, we have P [ X l 1 ∈ < ,...,B k 1 ,...,B k r ,X l ( n - r ) ∈ < ] = 1 · P ( B k 1 ) · ··· 1 · P ( B k r ) · 1 . Here, l j denotes the subscribe not equal to any k i . So random variables X 1 ,...,X n are independent if for any collection of sets A 1 ,...,A n , [ X 1 A 1 ] ,..., [ X n A n ] are independent events. We also learned before that the joint cdf captures all information about the the probability distribution of random vectors. So we should expect that there will be some criterion involving cdfs which will characterize indepen- dence of random variables. Random variables X 1 ,...,X n are independent if and only if we can write the joint cdf as the product of the marginal cdfs of the random variables, that is, F X 1 , ··· ,X n ( x 1 ,...,x n ) = F X 1 ( x 1 ) ··· F X n ( x n ) , ∀ - ∞ < x 1 ,...,x n < . This means that given the marginal distributions of the independent ran- dom variables, the joint distribution of the random vector generated by those random variables is determined. 1

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1 INDEPENDENT RANDOM VARIABLES Similarly, when X 1 ,...,X n are discrete, they are independent if and only if their joint pmf can be written as a product of their marginal pmfs, that is, p X 1 , ··· ,X n ( x i 1 ,...,x i n ) = p X 1 ( x i 1 ) ··· p X n ( x i n ) , x i 1 ,...,x i n . And the case is that of jointly distributed continuous random variables. In
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Lecture_9 - 1 INDEPENDENT RANDOM VARIABLES Lecture 9...

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