chap3 - Discrete random variables Chapter 3 excluding...

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Discrete random variables Chapter 3 excluding 3-7.2, 3-8 1
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Random variable A quantity whose value depends on the outcome of a random experiment • Airlines sell more tickets than the plane can hold: chance that every passenger who shows up can take the flight r.v. of interest: the number of passengers who show up 2
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A r.v. assigns a real number to each outcome of a random experiment X= sum of the numbers of a come-out throw {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1) } Ù X=7 A r.v. can not be multiple valued – Discrete : range is finite or countably infinite – Continuous : range consists of intervals of real numbers Denote r.v.’s by X, Y,…, their values by x, y,… 3
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Probability distribution For a come-out throw, interested in P(X=2), P(X=3), …, P(X=12) • Probability distribution : a description of the probabilities associated with possible values of X – Discrete r.v.: probability mass function ( pmf ) – Continuous r.v.: probability density function ( pdf ) Cumulative distribution function ( cdf ) Other ways to describe probability distribution 4
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Probability mass function of X= sum of numbers on two dice P(X=2) = 1/36 P(X=3) = 2/36 P(X=7) = 6/36 P(X=11) = 2/36 P(X=12) = 1/36 5
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Visualizing pmf 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 23456789 1 0 1 1 1 2 probability 6
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Probability mass function For discrete r.v. X with possible values x 1 ,…,x n , the pmf is defined by f(x i )=P(X=x i ). Key properties : Probabilities of events of interest craps out: P(X=2)+P(X=3)+P(X=12)=4/36 Throwing a natural: P(X=7)+P(X=11)=8/36 = = n i i i x f x f 1 1 ) ( , 0 ) ( 7
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Example: in a batch of 100 parts, 5 are defective. Two parts are randomly selected. X is the number of defective parts. What is the probability distribution of X ? P(X=0)=P(1 st is non-defective,2 nd is non-defective) =(95/100) x (94/99)=90.2% P(X=1)=P(1 st is defective, 2 nd is non-defective)+ P(1 st is non-defective,2 nd is defective) =(5/100) x (95/99)+(95/100) x (5/99)=9.6% P(X=2)=P(1 st is defective, 2 nd is defective)=? 8
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Cumulative distribution function •pm f specifies probability at each point: f(x i )=P(X=x i ) •cd f specifies probabilities of the form P(X x) for any real number x: Both of them describe a probability distribution: knowing one, can compute the other X= number of defective parts in previous example ) ( ) ( ) ( = = x x i i x f x X P x F 9
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Recall P(X=0)=90.2%, P(X=1)=9.6%, P(X=2)=0.2% F(-0.1)=P(X -0.1)= 0 F(x)=0 for any x<0 F(0)=P(X 0)= P(X=0) =90.2% ( jump @ 0, size P(X=0) ) F(0.5)=P(X 0.5 )=P(X=0)=90.2% F(x)=90.2% for any 0 x<1 F(1)=P(X 1 )= P(X=0)+P(X=1) =99.8% ( jump @ 1, size P(X=1) ) F(x)=99.8% for any 1 x<2 F(2)=P(X 2)= P(X=0)+P(X=1)+P(X=2) =1 ( jump @ 2, size P(X=2) ) F(X)=1 for any x 2 10
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Visualizing cdf 11
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