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851lecture11

# 851lecture11 - Statistics 851(Fall 2013 Prof Michael...

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Statistics 851 (Fall 2013) September 27, 2013 Prof. Michael Kozdron Lecture #11: Continuity of Probability (continued) We will now apply the continuity of probability theorem to prove that the function F ( x ) = P { ( −∞ , x ] } , x R , defined last lecture is actually a distribution function. Theorem 11.1. Consider the real numbers R with the Borel σ -algebra B , and let P be a probability on ( R , B ) . The function F : R [0 , 1] defined by F ( x ) = P { ( −∞ , x ] } for x R is a distribution function; that is, (i) lim x →−∞ F ( x ) = 0 and lim x →∞ F ( x ) = 1 , (ii) F is right-continuous, and (iii) F is increasing. Proof. In order to prove that F ( x ) = P { ( −∞ , x ] } is a distribution function we need to verify that the three conditions in definition are met. We will begin by showing (ii). Thus, to show that F is right-continuous, we must show that if x n is a sequence of real numbers converging to x from the right, i.e., x n x or x n x +, then F ( x n ) converges to F ( x ), i.e., lim x n x + F ( x n ) = F ( x ) . However, this follows immediately from the continuity of probability theorem (actually it follows directly from Exercise 10.3 which follows directly from Theorem 10.2) by noting that if x n x +, then ( −∞ , x 1 ] ( −∞ , x 2 ] · · · ( −∞ , x j ] ( −∞ , x j +1 ] · · · ( −∞ , x ] and n =1 ( −∞ , x n ] = ( −∞ , x ] so that lim x n x + F ( x n ) = lim n →∞ P { ( −∞ , x n ] } = P n =1 ( −∞ , x n ] = P { ( −∞ , x ] } = F ( x ) .

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