5 if x1 xn are independent then var x1 xn

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Unformatted text preview: o sets A and B we have P (X 2 A, Y 2 B ) = P (X 2 A)P (Y 2 B ) (A.13) In the discrete case, (A.13) is equivalent to P (X = x, Y = y ) = P (X = x)P (Y = y ) (A.14) for all x and y . The condition for independence is exactly the same in the continuous case: the joint distribution is the product of the marginal densities. fX,Y (x, y ) = fX (x)fY (y ) (A.15) The notions of independence extend in a straightforward way to n random variables: the joint probability or probability density is the product of the marginals. Two important consequences of independence are Theorem A.1. If X1 , . . . Xn are independent, then E (X1 · · · Xn ) = EX1 · · · EXn Theorem A.2. If X1 , . . . Xn are independent and n1 < . . . < nk n, then h1 (X1 , . . . Xn1 ), h2 (Xn1 +1 , . . . Xn2 ), . . . hk (Xnk 1 +1 , . . . X nk ) are independent. In words, the second result says that functions of disjoint sets of independent random variables are independent. Our last topic in this section is the distribution of X + Y w...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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