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Topic_4 - Topic 4 Continuous distributions Basics of...

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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  
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2 Continuous Random Variables 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 - = =
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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 =
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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 = =
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5 Requirements of a pdf A pdf must satisfy the following two requirements: Does the pipeline pdf satisfy these requirements? - = ( ) 0  for all   or ran ges of  ( ) 1 f x x x f x dx 1 0 100 ( ) 100 0 x f x otherwise = 100 100 0 0 1 100 100 x dx =
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6 Mean and variance of a cont. random variable μ σ μ σ μ μ - = = = - = - = = 2 2 2 2 2   ( ),  m ean  of   or expected valu e of    ( ( )) ( ) ( ) ,  expected valu e of  ( )   [( ) ],  varian ce of    ( )   ( ) ( ),  m om en t gen eratin g fu n ction  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
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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 - = = - 0 for 1 ( ) for for 1 x A x A x A F X dx A x B B A B A x B < - = = - -
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8 Uniform, etc. What is the mean of the Uniform distribution? What is the variance of the Uniform distribution? Using the same methodology as outlined above…. 1 B x A x dx B A μ = = - 2 B A + 2 ( ) E X = 2 2 3 B AB A + + 2 x σ = 2 ( ) 12 B A -
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9 Gamma distribution The  gamma distribution Γ(α,β29,  is defined by the following pdf  where  This is more for background purposes.  We will not be doing Gamma  calculations by hand.
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