This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Statistical Estimation Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign. All Rights Reserved ECE 313 Probability with Engineering Applications ECE 313  Lecture 9 © 2000 Dilip V. Sarwate, University of Illinois at UrbanaChampaign, 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 UrbanaChampaign, 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 ith 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 UrbanaChampaign, 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 UrbanaChampaign, 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 closedform 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 UrbanaChampaign, 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 UrbanaChampaign, 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)...
View
Full
Document
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.
 Spring '09
 mr.pil

Click to edit the document details