GE 331-Lecture 9

# GE 331-Lecture 9 - Cumulative Distribution Function The...

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IE 300/GE 331 Lecture 9 Negar Kiyavash, UIUC 1 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 9 Negar Kiyavash, UIUC 2 CDF (cont) 0.00202 2 0.09596 1 0.90202 0 f(x) x
IE 300/GE 331 Lecture 9 Negar Kiyavash, UIUC 3 Mean (cont) • The mean of a discrete random variable X • The variance of a discrete r.v. X ) ( X E = μ ) ( ) ( ) ( ) ( 1 2 2 1 1 = = + + + = n i i i n n x f x x f x x f x x f x L = = = = n i i i x f x X E X V 1 2 2 2 ) ( ) ( ) ( ) ( σ

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IE 300/GE 331 Lecture 9 Negar Kiyavash, UIUC 4 Recall: E(aX+b)=aE(X)+b (Expectation is linear) V(aX+b)=a 2 V(X) • Discrete unifrom distribution: (X can take n values) • Bernouli distribution (trial): (X takes two values) • Binomial distribution: (X can be any value 0 k n) p - 1 0) P(X , 1) P(X = = = = p n 0,1,. .., k , ) 1 ( k n k) P(X f(k) = = = = k n k p p n x f / 1 ) ( =
IE 300/GE 331 Lecture 9 Negar Kiyavash, UIUC 5 Mean & Variance for Binomial Distribution • For the k th Bernoulli trials, let X k =1 if success, X k =0 if failure, k=1,2,…,n – Recall that E(X k )=p, V(X k )=p(1-p) • Then the binomial r.v. X=X 1 +X 2 +…+X n • Mean and variance of binomial r.v.

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IE 300/GE 331 Lecture 9
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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 9 - Cumulative Distribution Function The...

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