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2303 DiscreteDistributions F10

2303 DiscreteDistributions F10 - DISCRETE PROBABILITY...

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DISCRETE PROBABILITY DISCRETE PROBABILITY DISTRIBUTIONS: DISTRIBUTIONS: The hyperGeometric, Binomial and The hyperGeometric, Binomial and Poisson models Poisson models
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A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes : number of complaints per day number of TV’s in a household number of rings before the phone is answered Only two possible outcomes : gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first Discrete Probability Discrete Probability Distributions Distributions
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Discrete Probability Discrete Probability Distributions Distributions Apply simple probability rules to develop equations for some simple situations Codify these as “known” probability distributions IMPORTANCE VERY common as approximations to real-world Used to build other statistical tools
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Uniform Probability Distributions Uniform Probability Distributions The uniform probability distribution uniform probability distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable P(x) m n If I’m a fair die, each probability (1,2,3,4,5,6) is the same: 1/6.
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Hypergeometric Distribution Hypergeometric Distribution N S n x Out of N items, S are “special”. I take a sample of n from the N items. What is the probability that x of those n are “special”?
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Hypergeometric Example 1 Hypergeometric Example 1 I know there are S=3 defective products in a batch of N=10. By mistake n=5 products from this batch were shipped to a customer. What is the chance the customer got x=2 defective products General formula: Prob(defect) depends on how many successes we have so far. This is called “sampling without replacement”. N n S N x n S x C C C x P - - × = ) (
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Combinations Combinations A combination combination is an outcome of an experiment where x objects are selected from a group of n objects.
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Combinations Combinations COUNTING RULE FOR COMBINATIONS COUNTING RULE FOR COMBINATIONS where n! = n(n - 1)(n - 2) . . . (2)(1) x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1 and x ≤ n )! ( ! ! ) , ( x n x n C x n C n x - = =
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Factorials and Combinations Factorials and Combinations 10 1 2 20 ! 3 ! 2 ! 3 4 5 )! 2 5 ( ! 2 ! 5 ) 2 , 5 ( 5 2 = × = × × × = - = = C C 24 1 2 3 4 ! 4 1 ! 0 = × × × = =
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Prob(2 defective products) = C 3 2 * C 10-3 5-2 / C 10 5 = (3!/2!1!) * (7!/3!4!) / (10!/5!5!) = 3 * 35 / 252 = 0.417 Hypergeometric Example 1 Hypergeometric Example 1 (cont.) (cont.)
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P(x) = Prob(x successes) =(# ways of choosing x out of S) * (# ways of choosing n-x out of N-S) / (# ways of choosing n out of N) = C S x * C N-S n-x / C N n Hypergeometric Distribution Hypergeometric Distribution
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E(X) = np (where p = S/N) Var(X) = np(1-p)(N-n)/(N-1) Hypergeometric Distribution Hypergeometric Distribution
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Hypergeometric Distribution Hypergeometric Distribution Example 2 Example 2 Using an experimental procedure 7 out of 12 cancer patients have been cured.
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