Topic_4 - 1 Topic 4 - Continuous distributions Basics of...

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Unformatted text preview: 1 Topic 4 - Continuous distributions Basics of continuous distributions - pages 119 - 124 Uniform distribution - pages 135 136 Normal distribution - pages 125 - 131 Gamma distribution - pages 138 - 141 2 Continuous Random Variables A continuous random variable can take on values from an entire interval of the real line. The probability density function (pdf) of a continuous random variable, X , is a function f ( x ) such that for a < b The cumulative density function (cdf) of X is defined as ( ) ( ) b a P a X b f x dx = ( ) ( ) ( ) x F x P X x f t dt- = = 2 3 Some relationships What is the relationship between the pdf ( f) and the cdf ( F) ? You integrate the pdf to get the cdf You take the derivative of the cdf to get the pdf P ( a X b ) = F ( b ) F ( a ) P(X = a) = P ( a X a ) = F ( a ) F ( a ) = 0 ' ( ) ( ) f x F x = 4 Pipeline example A pipeline is 100 miles long and every location along the pipeline is equally likely to break Let X be the distance measured in miles from the pipeline origin where a break occurs What is the cdf for X ? What is the pdf for X ? What is P(30 X 50)? ( ) , for 0 100 100 x P X x x = ' 1 ( ) ( ) , for 0 x 100, 0 otherwise 100 f x F x = = 5 Requirements of a pdf A pdf must satisfy the following two requirements: Does the pipeline pdf satisfy these requirements?- = ( ) 0 for all or ranges of ( ) 1 f x x x f x dx 1 100 ( ) 100 x f x otherwise = 100 100 1 100 100 x dx = 6 Mean and variance of a cont. random variable - = = = - = - = = 2 2 2 2 2 ( ), mean of or expected value of ( ( )) ( ) ( ) , expected value of ( ) [( ) ], variance of ( ) ( ) ( ), moment generating function for (0) , X X X X X tX X X X E X X X E h X h x f x dx h X E X X E X M t E e X M = 2 (0) ( ) X M E X 7 Uniform distribution A uniform distribution on the interval from A to B , U ( A , B ), is defined by a pdf of the form Does f ( x ) meet requirements? What is the cdf for the Uniform distribution? 1 ( ) for f x A x B B A =- 1 all ( ) 0 and 1- B A B A f x dx B A B A- = =- for 1 ( ) for for 1 x A x A x A F X dx A x B B A B A x B <- = =-- 8 Uniform, etc. What is the mean of the Uniform distribution?...
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This note was uploaded on 03/19/2012 for the course MATH MATH 304 taught by Professor Young during the Spring '09 term at Texas A&M.

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Topic_4 - 1 Topic 4 - Continuous distributions Basics of...

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