Chapter 5

Chapter 5 - IMSE 213: Probability and Statistics for...

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Click to edit Master subtitle style IMSE 213: Probability and Statistics for Engineers Instructor: Steven E. Guffey, PhD, CIH © 2002-2009 Chapter 5: Some Discrete Probability Distributions
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22 Introduction Will cover most common discrete probability distributions Simplest distribution: Equal probability distribution I.e., all events equally likely If there are k values with equal probability, then: Examples: Toin toss Random selection from a group
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33 Theorem 5.1: Mean and Variance of Discrete Uniform Distribution ) ( 1 i f x k i i = = μ ( 29 k x k i i 2 1 2 = - = σ k x k i i = = 1 f(i) = n/N = 1/k
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44 5.3 Binomial Distribution Bernoulli Process: Experiment consists of n repeated trials Each trial outcome can be classified as “success” or “failure” (i.e., only 2 possible outcomes) Probability of success, p , remains constant from trial to trial The repeated trials are independent Number of successes, X, is called a binomial random variable b(x; n, p) Its distribution is called the binomial distribution
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55 Binomial Distribution Each success has probability, p Each failure has probability, q q = 1 - p x = no. successes n - x = no. failures Since independent, combination probabilities are multiplied: Prob of specific order = px qn-x Total number of sample points that have n trials and x successes and n-x failures is
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66 Example 5.4 The prob that a component will survive a given shock test is ¾. Find the probability that exactly 2 of the next 4 components will survive. Solution: Assuming the tests are independent And p = ¾ for each of the 4 tests, then:
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77 Sum of Probabilites Must Equal Unity As with all other distributions, the sum of all probabilities must equal unity:
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88 Table A.1: Binomial Cumulative Distribution Trials Success 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.9000 0.8000 0.7500 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 1 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0 0.8100 0.6400 0.5625 0.4900 0.3600 0.2500 0.1600 0.0900 0.0400 0.0100 2 1 0.9900 0.9600 0.9375 0.9100 0.8400 0.7500 0.6400 0.5100 0.3600 0.1900 2 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3 0 0.7290 0.5120 0.4219 0.3430 0.2160 0.1250 0.0640 0.0270 0.0080 0.0010 3 1 0.9720 0.8960 0.8438 0.7840 0.6480 0.5000 0.3520 0.2160 0.1040 0.0280 3 2 0.9990 0.9920 0.9844 0.9730 0.9360 0.8750 0.7840 0.6570 0.4880 0.2710 3 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 4 0 0.6561 0.4096 0.3164 0.2401 0.1296 0.0625 0.0256 0.0081 0.0016 0.0001 4 1 0.9477 0.8192 0.7383 0.6517 0.4752 0.3125 0.1792 0.0837 0.0272 0.0037 4 2 0.9963 0.9728 0.9492 0.9163 0.8208 0.6875 0.5248 0.3483 0.1808 0.0523 4 3 0.9999 0.9984 0.9961 0.9919 0.9744 0.9375 0.8704 0.7599 0.5904 0.3439 4 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Proportion, p
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99 Example 5.5 The prob that a patient recovers from a disease is p= 0.4 . For n= 15 with the disease, what is the probability: A) at least 10 survive? Trials Success 0.1 0.2 0.25 0.3 0.4 0.5 15 5 0.9978 0.9389 0.8516 0.7216 0.4032 0.1509 15 6 0.9997 0.9819 0.9434 0.8689 0.6098 0.3036 15 7 1.0000 0.9958 0.9827 0.9500 0.7869 0.5000 15 8 1.0000 0.9992 0.9958 0.9848 0.9050 0.6964 15 9 1.0000 0.9999 0.9992 0.9963 0.9662 0.8491 15 10 1.0000 1.0000 0.9999 0.9993 0.9907 0.9408 P(X > 10 ) = 1- P(X < 10 ) = = - = x b ) , , ( 1 x 15 0.4 1- 0.9662 = 0.0338 0 9
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1010 Example 5.5b The prob that a patient recovers from a rare blood disease is 0.4 . What is the probability that, of 15, P(3 < X < 8) Trials Success 0.1 0.2 0.25 0.3 0.4 15 2 0.8159 0.3980 0.2361 0.1268 0.0271 15 3 0.9444 0.6482 0.4613 0.2969 0.0905 15 4 0.9873 0.8358 0.6865 0.5155 0.2173 15 6 0.9997 0.9819 0.9434 0.8689 0.6098 15 8 1.0000 0.9992 0.9958 0.9848 0.9050 8 2 0 0 ( ;15,0.4) ( ;15,0.4) x x b x b x = = = - 0.9050 0.0271 0.8779 = - = ( ; , ) x b x p = = 3 8 15 0.4
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Example 5.5c The probability that a patient recovers from a rare blood disease is 0.4 . What is the probability that: C) exactly 5 of 15 survive ?
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Chapter 5 - IMSE 213: Probability and Statistics for...

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