ECE313.Lecture09

ECE313.Lecture09 - Statistical Estimation Professor Dilip...

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Unformatted text preview: Statistical Estimation Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 Review — Independent Trials Independent trials: the outcomes of the various trials do not influence or affect one another in any way Independence of trials is a belief and cannot be proved mathematically Compound experiment = independent trials of a simple experiment Simple versus compound events P(A, B, C, A c , …) = P(A)P(B)P(C)P(A c )… ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 Review — Random Variables X is a random variable defned on the simple experiment. X i is the value oF X on i-th sub experiment ( X 1 , X 2 , X 3 ,… ) is called a random vector ¡or repeated independent trials, the random variables X 1 , X 2 , X 3 , X 4 ,… are said to be independent random variables P( X 1 = a 5 , X 2 = a 2 , X 3 = a 7 , X 4 = a 9 , … ) = = P( X 1 =a 5 )P( X 2 =a 2 )P( X 3 =a 7 )P( X 4 =a 9 )… ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 Binomial Random Variables A binomial random variable Y with parameters (n, p) is the number of times an event A of probability p occurs on n independent trials Y takes on values 0, 1, 2, … n For 0 ≤ k ≤ n, p Y (k) = P{ Y = k} = p k (1 – p) n–k n k P(A occurred on a speci¡c set of k trials) = p k (1 – p) n–k ; P{ Y = k} is the probability that A occurred on some set of k trials ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 Probabilities from the binomial pmf Probabilities such as P{a < Y < b} are found by summing up the appropriate terms in the pmf There are no closed-form expressions for such probabilities: numerical evaluation is required P{a < Y < b} is the sum of L (say) of the n+1 probabilities in the pmf If L > (n+1)/2, Fnd 1 – P{complement} ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 Mean of binomial random variable E[ Y ] = np E[ Y ] = ∑ k• p k (1 – p) n–k n k=0 n k = ∑ k• p k (1 – p) n–k n k=1 n k = np• ∑ p k–1 (1 – p) (n–1)–(k–1) n k=1 n–1 k–1 E[ Y ] = np•(p + 1 – p) n–1 = np ECE 313 - Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 36 Variance of binomial RV var( Y ) = np(1–p) E[ Y ( Y –1)] = E[ Y 2 ] – E[ Y ] E[ Y ( Y –1)] = ∑ k(k–1)• p k (1 – p) n–k n k=0 n k = n(n–1)p 2 ∑ p k–2 (1 – p) (n–2)–(k–2)...
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This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture09 - Statistical Estimation Professor Dilip...

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