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Week 4 Mon Sept 20 - WEEK 4 Monday Sept 20 Today Cover...

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1 WEEK 4, Monday , Sept 20 Today: Cover Sections 3-1 through 3-3 . Discuss discrete and continuous random variables and their distributions as probability models. Outcomes are on the real number line, often the events of interest are intervals. Discrete Probability mass function . Properties are p ( x ) where p ( x ) ≥ 0 (3-1) and 1 ) ( x all x p . (3-2) Cumulative distribution function . x i i p x X P x F ) ( ) ( ) ( (3-3) Expectation . ) ( ) ( all x x xp X E (3-4) is called the mean of X or simply the expected value of X . For any function h and Y = h ( X ), Y is a random variable with expectation x all x p x h X h E Y E ) ( ) ( )) ( ( ) ( . (3-5) Variance and Standard Deviation . )] ( [ ) ( ] ) [( ) ( 2 2 2 2 2 2 x p x X E X E X V (3-7) and (3-8) ) ( ) ( X V X SD (3-9)
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2 Please note . The units of measure for E ( X ) and SD ( X ) are the same as the units of measure for X . If X is measured in feet , then E ( X ) and SD ( X ) are in units of feet . If X is in units of ($1,000), then E ( X ) and SD ( X ) are in units of ($1,000). Example 1. Total on the roll of a pair of fair dice . x 2 3 4 5 6 7 8 9 10 11 12 p ( x ) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Probability Mass Function p and Cumulative Distribution Function F x p (x) F ( x ) 2 1/36 1/36 3 2/36 3/36 4 3/36 6/36 5 4/36 10/36 6 5/36 15/36 7 6/36 21/36 8 5/36 26/36 9 4/36 30/36 10 3/36 33/36 11 2/36 35/36 12 1/36 36/36 Prob Dist of X = Total on Two Dice 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 10 11 12 x P(x)
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3 Prob Dist of X = Total on Two Dice 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 10 11 12 x P(x) Way to Calculate Mean, Variance, and Standard Deviation of a Discrete Random Variable 7 ) ( ) ( x xp X E 8333 . 5 36 / 210 7 ) 36 / 1974 ( ) ( ] ) [( ) ( 2 2 2 2 2 X E X
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