Lecture_17 - 1 UNIFORM AND BETA DISTRIBUTIONS Lecture 17...

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1 UNIFORM AND BETA DISTRIBUTIONS Lecture 17 ORIE3500/5500 Summer2009 Chen Class Today Uniform and Beta Distribution Poisson Distribution 1 Uniform and Beta distributions 1.1 Uniform Distribution The uniform distribution is probably the simplest of all continuous distrib- utions. A random variable X is said to be uniformly distributed between a and b , U ( a,b ), if picking any point between a and b is equally likely. Now there are uncountably many points between a and b , this means that the density is constant in the given range, that is f X ( x ) = 1 b - a ,a x b. The U ( a,b ) random variable has cdf F X ( x ) = 0 x < a x - a b - a a x < b 1 x b. One can compute the mean and variance of a U ( a,b ) random variable pretty easily: E ( X ) = Z b a x b - a dx = x 2 2( b - a ) b a = b + a 2 . E ( X 2 ) = Z b a x 2 b - a dx = x 3 3( b - a ) b a = b 3 - a 3 3( b - a ) = b 2 + ab + a 2 3 , and, var ( X ) = b 2 + ab + a 2 3 - b + a 2 · 2 = ( b - a ) 2 12 . 1
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1.1 Uniform Distribution 1 UNIFORM AND BETA DISTRIBUTIONS The mgf of a U ( a,b ) random variable is given by φ X ( t ) = Z b a e tx 1 b - a dx = ( e tb - e ta t ( b - a ) t 6 = 0 1 t = 0 . By L’Hospital Rule,
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell University (Engineering School).

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Lecture_17 - 1 UNIFORM AND BETA DISTRIBUTIONS Lecture 17...

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