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Unformatted text preview: Spring 2010 Pstat160b HO #1 Continuous Distribution summary Recall that if X is a continuous random variable, P ( X = x ) = 0, for any number x . Cumulative Distribution Function The C.d.f (cumulative distribution function) of a continuous random variable X , is a continuous function (of x ) F X ( x ) = P ( X x ) which satisfies the following requirements: F X ( x ) 1 F X ( x ) is an increasing function lim x  F X ( x ) = 0 lim x + F X ( x ) = 1 P ( a X b ) = P ( a < X < b ) = F X ( b ) F X ( a ) Probability Distribution Functions The P.d.f. (probability distribution function) is not necessary continuous but satisfies f X ( x ) P ( a X b ) = P ( a < X < b ) = Z b a f X ( x ) dx Z  f X ( x ) dx = 1 F X ( x ) = Z x f X ( u ) du The main continuous distributions that we are going to encounter are the Uniform, Exponential, Gamma , and Normal distributions. Uniform: U ( a,b ) , a < b f X ( x ) = 1 b a a x b otherwise Exponential: E ( ) , > f X ( x ) = e x x otherwise Gamma: G ( k, ) , k, > f X ( x ) = k ( k ) x k 1 e x x otherwise Normal: N ( , ) , > f X ( x ) = 1 2 e ( x ) 2 2 2 Beta: (first kind) B ( , ) , , > f X ( x ) = 1 B ( , ) x  1 (1 x )  1 x 1 otherwise 2 Figure 1. Expential density function for = 1 and = 3. Figure 2. Gamma density function for G ( . 5 , 2) , G (2 , 1) and G (5 ,. 5). Figure 3. Beta density function for B ( , 4 , 4) , B (2 . 5 , 2 . 5) B (5 , 5 and B (3 , 2). For the Gamma and Beta distribution: The symbol ( k ) and B ( r,s ) means respectively ( k ) = Z u k 1 e u du, ( k ) = ( k 1)! for integers k B ( , ) = Z 1 x  1 (1 x )  1 dx Note also: G (1 , ) = E ( ) G ( k/ 2 , 1 / 2) = 2 ( k ) (Chi square distribution with k degrees of freedom) If X 1 ,...,X n are n independent E ( ), then Y = n i =1 X i is G ( n, ). 3 Expected Values For a continuous random variable, the mean or expected value , often denoted by , is given, if it exists, by = E ( X ) = Z  xf X...
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This note was uploaded on 01/10/2011 for the course STAT PStat 160b taught by Professor Bonnet during the Spring '10 term at UCSB.
 Spring '10
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