Chapter_03_random_variables

Chapter_03_random_variables - Definition of a Random...

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Definition of a Random Variable ± Random Variable [m-w.org] : a variable that is itself a function of the result of a statistical experiment in which each outcome has a definite probability of occurrence Copyright © Syed Ali Khayam 2009 3
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Definition of a Random Variable ± A random variable is a mapping from an outcome s of a random experiment to a real number : X XS S →⊂ \ domain range called the age f S X is called the image of X Copyright © Syed Ali Khayam 2009 4
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Definition of a Random Variable S S : X XS \ head Random Experiment Sample Space Random Variable X(s) tail 01 R S x Copyright © Syed Ali Khayam 2009 5
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Definition of a Random Variable : XS S →⊂ \ X ( s ) X 1 2 R 3 4 5 6 Random xperiment Experiment Sample Space, S Random Variable S x Copyright © Syed Ali Khayam 2009 Image courtesy of www.buzzle.com/ 6
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Definition of a Random Variable ± More than one outcomes can be mapped to the same real number S S X(s) : X XS \ Random 0 1 S Experiment Sample Space Random Variable Copyright © Syed Ali Khayam 2009 Image courtesy of www.buzzle.com/ 7
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Types of Random Variables ± Discrete random variables: have a countable (finite or infinite) image Ö S x = {0, 1} Ö S x = {…, -3, -2, -1, 0, 1, 2, 3, …} ± Continuous random variables: have an uncountable image Ö S x = (0, 1] Ö S x = R ± Mixed random variables: have an image which contains continuous and discrete parts Ö S x = {0} U (0, 1] ± We will mostly focus on discrete and continuous random variables Copyright © Syed Ali Khayam 2009 8
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Discrete Random Variables Copyright © Syed Ali Khayam 2009 9
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Probability Mass Function ± The Probability Mass Function ( pmf ) or the discrete probability density function provides the probability of a particular point in e sample space of a discrete random variable (rv) the sample space of a discrete random variable (rv) ± For a countable S X ={ a 0 , a 2 , …, a n }, the pmf is the set of probabilities ( ) {} Pr , 1,2, , Xk k pa X a k n == = p X ( a k ) S X ={ a 0 =0, a 1 =1, …, a 5 =5}, Copyright © Syed Ali Khayam 2009 X 0 1 2 3 4 5 10
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Properties of a PMF ± P1: ( ) 01 Xk pa ≤≤ ± P2: ( ) 1 kX aS = p X ( a k ) S X ={ a 0 =0, a 1 =1, …, a 5 =5}, Copyright © Syed Ali Khayam 2009 X 0 1 2 3 4 5 11
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Cumulative Distribution Function (CDF) of Discrete Random Variable ± The Cumulative Distribution Function ( CDF ) for a discrete rv is defined as: ( ) {} () Pr XX xt Ft X t p x =≤ = p X ( x ) mf F X ( x ) pmf CDF X 12 3 4 X 1 2 3 4 Copyright © Syed Ali Khayam 2009 12
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Cumulative Distribution Function (CDF) of Discrete Random Variable ± CDF can be used to find the probability of a range of values in a rv’s image: {} { } { } ( ) () Pr X aXb Xb Xa Fb Fa <≤= ≤− =− XX p X ( x ) F X ( x ) pmf CDF Copyright © Syed Ali Khayam 2009 X 12 3 4 X 3 4 13
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