Lecture 05-discrete probability distributions

Lecture 05-discrete probability distributions - 1036:...

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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 5 Lecture 5 Some Discrete Some Discrete Probability Distributions Probability Distributions
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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-2 Discrete Uniform Distribution • If a Random variable X have values of x 1 , x 2 ,… x k , with equal probabilities , then the discrete uniform distribution is given by • the discrete uniform distribution depends on the parameter k . • The mean and variance of the discrete uniform distribution f ( x ; k ) are k x x x x k k x f , , , , 1 ) ; ( 2 1 K = = k x k i i = = 1 µ () k x k i i = = 1 2 2 σ
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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-3 Bernoulli Process • The experiment consists of n repeated trials • Each trial results in an outcome that may be classified as a success or a failure (2 possible outcomes) • The probability of success, denoted by p , remains constant from trial to trial • The repeated trials are independent •T h e n u m b e r X of successes in n Bernoulli trials is called a binomial random variable . • The probability distribution of the binomial random variable X is n x q p q p x n p n x b x n x , , 2 , 1 , 0 , 1 where , ) , ; ( K = = + = Identically, independently, distribution (i.i.d.)
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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-4 Binomial Distribution • The binomial distribution derives it name form the fact that the n +1 terms in the binomial expansion of ( p +q) n correspond to the various values of b ( x ; n, p ) for x =0,1,2,…, n . That is, • For simplicity, we define 1 ) , ; ( ) , ; 2 ( ) , ; 1 ( ) , ; 0 ( 2 1 0 ) ( 2 2 1 = + + + + = + + + + = + p n n b p n b p n b p n b p n n q p n pq n q n q p n n n n n L L = = r x p n x b p n r B 0 ) , ; ( ) , ; (
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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-5 Example 5.5 • The probability that a patient recovers from a rare blood disease is 0.4. If 15 persons infected, then the probability of (a). at least 10 recovery, (b). from 3 to 8 survive, and (c). exactly 5 survive? Sol: Let X be the number of persons that recover ()( ) ) 4 0 15 9 ( 1 4 . 0 , 15 ; 10 . 15 10 . , ; B x b X P a x = = = () ( ) ) 4 . 0 , 15 ; 2 ( ) 4 . 0 , 15 ; 8 ( 4 . 0 , 15 ; 8 3 . 8 3 B B x b X P b x = = = ) 4 . 0 , 15 ; 4 ( ) 4 . 0 , 15 ; 5 ( ) 4 . 0 , 15 ; 5 ( 5 . B B b X P c = = =
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Prob. & Stat. Lecture05 - discrete probability distribution (cwliu@twins.ee.nctu.edu.tw) 5-6 Example 5.6 • A retailer purchases an electronic device with a defective rate of 3%. The inspector randomly picks 20 items. What is the probability of at least 1 defective item among these 20? • Suppose that the retailer receives 10 shipments in a month and the inspector randomly tests 20 devices per shipment.
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This note was uploaded on 08/23/2009 for the course IEE 1036 taught by Professor Cwliu during the Spring '06 term at National Chiao Tung University.

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Lecture 05-discrete probability distributions - 1036:...

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