GE 331-Lecture 11 - Continuous Random Variables IE 300/GE...

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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 1 Continuous Random Variables
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 2 Continuous Random Variables PDF: counterpart of probability mass function for a continuous random variable For a continuous random variable, P(X=x)=0 for any possible value x PDF: f(x)
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 3 Probability Density Function For a continuous r.v., probability that X falls into an interval [a, b] is: f(x) is called probability density function (pdf) The area under f(x) between a and b ) ( ) ( = b a dx x f b X a P b a dx x f ) (
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 4 PDF (cont) Properties: f(x) 0 because probabilities are always non-negative because P( −∞ X ≤∞ )=1 Because the probability of any point is zero: 1 ) ( = +∞ dx x f = < < = < = < = b a dx x f b X a P b X a P b X a P b X a P ) ( ) ( ) ( ) ( ) (
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 5 Uniform distribution (cont) 0 1 x f(x) 1 pdf for U [0, 1] E[U]=1/2 and Var[U]=1/12
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 6 Uniform distribution (cont) More generally, let X ~ U(0,1), then Y=(b-a)X+a ~ U(a,b) Mean : Variance :
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 7 Uniform distribution (cont) More generally, let X ~ U(0,1), then Y=(b-a)X+a ~ U(a,b) Mean : E(Y)=(b-a)E(X)+a=(b+a)/2 Variance : V(Y)=(b-a) 2 V(X)=(b-a) 2 /12 cdf : In Excel/Matlab, rand() generates U[0,1] Uniform distribution plays an important role in generating more complicated random numbers b y a a b a y dx a b y Y P y F y a = = = for 1 ) ( ) (
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IE 300/GE 331 Lecture 11 Negar Kiyavash, UIUC 8 Example Suppose that the probability density function of some contamination particle size (in micrometers) is known to be What is the probability that the particle size is between 2 and 3 micrometers What is the cumulative distribution function What is the expected size of the particle > = 1 0 1 / ) ( 3 x x x c x f
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This note was uploaded on 09/08/2009 for the course GE 331 taught by Professor Negarkayavash during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE 331-Lecture 11 - Continuous Random Variables IE 300/GE...

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