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Lecture 9 - Review of Independece Random Vectors

# Lecture 9 - Review of Independece Random Vectors - 1 REVIEW...

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1 REVIEW OF INDEPENDENCE RANDOM VECTORS Lecture 9 ORIE3500/5500 Summer2011 Li 1 Review of Independence Random Vectors Recall that X 1 , . . . , X n are said to be independent random variables if for any sets A 1 , . . . , A n and each A j ⊂ ℜ , j = 1 , . . . , n : P [ X 1 A 1 , . . . , X n A n ] = P [ X 1 A 1 ] ··· P [ X n A n ] . In particular, X and Y are independent discrete random variables if P ( X = x, Y = y ) = P ( X = x ) P ( Y = y ); X and Y are independent continuous random variables if f X,Y ( x, y ) = f X ( x ) f Y ( y ). Examples that use the deﬁnition of independence: Example A man and a woman decide to meet at a certain location. If each of them independently arrives at a time uniformly distributed between 12 noon and 1pm. Find the probability that the ﬁrst to arrive has to wait longer than 10 minutes. Solution: Let X and Y denote the time past 12 that the man and the woman arrive. Then X and Y are independent random variables, each of which is uniformly distributed over (0 , 60). Hence the answer is 2 P ( X +10 < Y ) = 2 ∫ ∫ x +10 <y f ( x, y ) dxdy = 2 60 10 y - 10 0 (1 / 60) 2 dxdy = 25 / 36 . Examples to check whether two random variables are independent or not: Example If ( X, Y ) are jointly distributed with joint density f X,Y ( x, y ) = 6 xy 2 , 0 < x, y < 1 , then are X and Y independent? To check this we would need to ﬁnd the marginals of X and Y ﬁrst. We do that by integrating over the other variable. f X ( x ) = 1 0 6 xy 2 dy = 2 x, 0 < x < 1 , and 1

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1 REVIEW OF INDEPENDENCE RANDOM VECTORS f Y ( y ) = 1 0 6 xy 2 dx = 3 y 2 , 0 < y < 1 . To see if the two random variables are independent or not we need to check if the joint density is the product of the two marginals. Indeed we get f X,Y ( x, y ) = f X ( x ) f Y ( y ) , and hence X and Y are independent. Example Let us consider a little variation of the previous example. If ( X, Y ) have joint pdf f X,Y ( x, y ) = 15 xy 2 , 0 < y < x < 1 . Here although the part 15
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Lecture 9 - Review of Independece Random Vectors - 1 REVIEW...

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