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
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.
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 - - × = ) (
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 - = =
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.)
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
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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## This note was uploaded on 02/13/2011 for the course ADM na taught by Professor Na during the Winter '09 term at University of Ottawa.

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2303 DiscreteDistributions F10 - DISCRETE PROBABILITY...

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