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 , de±ned 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] deFned by F ( x P { ( −∞ ] } 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 { ( −∞ ] } is a distribution function we need to verify that the three conditions in de±nition 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 )convergesto 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 ( −∞ 1 ] ( −∞ 2 ] ⊇···⊇ ( −∞ j ] ( −∞ j +1 ] ( −∞ ] and ° n =1 ( −∞ n ]=( −∞ ] so that lim x n x + F ( x n )= l im n →∞ P { ( −∞ n ] } = P ± ° n =1 ( −∞ n ] ² = P { ( −∞ ] } = F ( x ) .
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851lecture11 - Statistics 851(Fall 2013 Prof Michael...

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