JMF-4-07 - Discrete Random Variables Chapter 4 Notes Set 4...

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CH Reilly UCF - IEMS - STA 3032 1 Discrete Random Variables Chapter 4 Notes Set 4 Spring 2007
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CH Reilly UCF - IEMS - STA 3032 2 Random variable A random variable is a function that assigns an (unknown) numerical value from a sample space to each possible outcome or trial. Discrete random variable (Chapter 4) – countable number of possible values. Continuous random variable (Chapter 5) – uncountable number of possible values. The probability distribution for a random variable may be represented in functional, tabular, or graphical form.
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CH Reilly UCF - IEMS - STA 3032 3 Probability distribution X is a random variable whose realized value is denoted by x . (Other letters, besides X and x , will sometimes be used to represent random variables and their values.) The probability distribution of X tells us f(x)= Pr (X=x) for all possible values x of X .
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CH Reilly UCF - IEMS - STA 3032 4 Probability distribution (cont’d.) y.) probabilit of rules the (Recall 1 ) ( outcome.) simple a is ( . , 0 ) Pr( ) ( all = 2200 = = x x f x x x X x f
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CH Reilly UCF - IEMS - STA 3032 5 Probability distribution (cont’d.) There are a number of ways to represent a probability distribution for a discrete random variable graphically, including: Probability histogram – total area is 1 square unit. Probability bar chart – heights of “pegs” represent the probabilities for each possible value. Sometimes the probability distribution function for a discrete random variable is called a probability mass function (pmf).
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CH Reilly UCF - IEMS - STA 3032 6 Cumulative distribution function F(x) = Pr( X x ) for all possible values x of X . The cumulative distribution function is often referred to as a cdf for convenience; sometimes it is simply called a distribution function. Note that differences in cdf values can be used to find probabilities for particular values of X .
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CH Reilly UCF - IEMS - STA 3032 7 Expected value of a random variable The expected value of a random variable is the mean value of the random variable. Expected values are population parameters. Expected values are sometimes called expectations. (I use the terms interchangeably.) The expectation of a random variable is not necessarily one of the possible values of the random variable.
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CH Reilly UCF - IEMS - STA 3032 8 Expected value (cont’d.) = = = = x x x x f x X x f x g X g x f x X all 2 2 all all ) ( ) ( E ) ( ) ( )) ( ( E ) ( ) ( E μ
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CH Reilly UCF - IEMS - STA 3032 9 Variance of X ( 29 ( 29 ( 29 2 2 2 2 2 2 2 ) ( E ) ( E ) ( E E ) ( Var σ μ + = - = - = - = = X X X X X
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CH Reilly UCF - IEMS - STA 3032 10 Example x 1 3 4 6 f(x) 0.10 0.20 0.40 0.30 F(x) 0.10 0.30 0.70 1.00
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CH Reilly UCF - IEMS - STA 3032 11 Example (cont’d.) ( 29 5133 . 1 29 . 2 ) 1 . 4 ( 1 . 19 ) ( E ) ( E 1 . 19 ) 30 . 0 )( 6 ( ) 10 . 0 )( 1 ( ) ( E 1 . 4 ) 30 . 0 )( 6 ( ) 10 . 0 )( 1 ( ) ( E 2 2 2 2 2 2 2 = - = - = = + + = = + + = = σ μ X X X X
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This note was uploaded on 08/22/2010 for the course STA 3032 taught by Professor Sapkota during the Spring '08 term at University of Central Florida.

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JMF-4-07 - Discrete Random Variables Chapter 4 Notes Set 4...

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