GE 331-Lecture 8

GE 331-Lecture 8 - Discrete Random Variables IE 300/GE 331...

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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 1 Discrete Random Variables
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 2 Motivation (cont) sample space Real number line Example A coin is flipped three times. The sample space for this experiment is ={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Let random variable X be the number of heads in three coin tosses. • X assigns each outcome in a number from the set {0, 1, 2, 3}. HHH HHT HTH HTT THH THT TTH TTT 32212110 There is nothing random about the Mapping!
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 3 Cumulative Distribution Function •T h e cumulative distribution function (cdf) of X • Well defined for any real number x ) ( ) ( ) ( = = x x i i x f x X P x F
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 4 CDF Example: F(-0.1)= F(0)= F(0.5)= F(1)= X012 f(X) 0.90202 0.09596 0.00202
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 5 CDF Example: F(-0.1)=P(X -0.1)=0, F(x)=0 for any x<0 F(0)=P(X 0)=P(X=0)=0.90202 F(0.5)=P(X 0.5 )=P(X=0)=0.90202=F(x) for any 0 x<1 F(1)=P(X 1 )=P(X=0)+P(X=1)=0.99798 X012 f(X) 0.90202 0.09596 0.00202
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 6 CDF (cont) For discrete r.v., cdf is a step function with jumps at x i , the jump size: P(X=xi)=f(xi)
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 7 CDF (cont) •T h e c d f o f X (the number of defects a parts can have -- up to 2 defects) – Jumps at x=0,1,2 f(0)=0.90202, f(1)=0.09596, f(2)=0.00202 < < < = 2 1 2 1 99798 . 0 1 0 90202 . 0 0 0 ) ( x x x x x F
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 8 Mean (expected value) • Location (center) of a probability distribution • Wrong: (0+1+2)/3=1 • Probability weighted average : 0*0.90202+1*0.09596+2*0.00202=0.1 xf ( x ) 0 0.90202 1 0.09596 2 0.00202
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IE 300/GE 331 Lecture 8 Negar Kiyavash, UIUC 9 Mean (cont)
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GE 331-Lecture 8 - Discrete Random Variables IE 300/GE 331...

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