X p x x e x μ 2 2 2 x p x e x x e x var x x μ σ 26

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) ( * ] [ x P x X E X = = μ ) ( )] ( [ ] ) [( ] [ 2 2 2 x P X E x X E X Var X X - = - = = μ σ
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26 Discrete Probability Distributions Binomial distribution: A trial has only two possible outcomes – “success” or “failure” There is a fixed number, n (finite), of identical trials The trials of the experiment are independent of each other The probability of a success, p, remains constant from trial to trial If p represents the probability of a success, then, (1- p ) = q is the probability of a failure
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27 Discrete Probability Distributions P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, ( x = 0, 1, 2, ..., n ) n = random sample size p = probability of a success, q = probability of a failure ( q = 1- p ) n ! = n ( n -1)( n -2)……1 and 0!= 1 (by definition) Probability mass function: Mean: Variance and Standard Deviation: X = E[X] = 2= Var[X] = np npq σ = P ( x ) = P ( X = x ) = n x ÷ p x q n - x = n ! x !( n - x )! p x q n - x npq
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Discrete Probability Distributions 28 Pois s on Dis tribution: Useful in estimating the number of occurrences over a specified interval of time or space . Properties and as s umptions of the Pois s on dis tribution Can take any non-negative integer value (x=0,1,2,3, …… .) : the average number of outcomes of interest per unit time segment or unit space segment. The probability that an outcome of interest occurs in a given (time) segment is the same for all segments of interest of the same length. What happens in one segment is independent from what happens in any other segment.
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Discrete Probability Distributions 29 Pois s on Dis tribution Probability mass function: Expectation (Mean) x = number of successes in segment of interest  = average rate of occurrence per unit segment (e.g., #of calls per hour) t = size of the segment of interest (e.g., a three-hour interval) e = base of the natural logarithm system (2.71828...) Variance Standard Deviation ! ) ( ) ( x e t x X P t x λ λ - = = λt μ = λt σ 2 = = σ t λ
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Example 30 Once a touristic town, Wonderland does not attract visitors any more. A small gift shop located in Wonderland is occasionally visited by customers. Customers arrive one at a time (no couples or groups). Male customers end up purchasing a gift with a probability of 60%. Female customers, on the other hand, end up purchasing a gift with a probability of 80%. Also, 65% of all customers are female. Customers make purchasing decisions independently of each other. a) What is the probability that the next customer will purchase a gift?
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Example 31 Once a touristic town, Wonderland does not attract visitors any more. A small gift shop located in Wonderland is occasionally visited by customers. Customers arrive one at a time (no couples or groups). Male customers end up purchasing a gift with a probability of 60%. Female customers, on the other hand, end up purchasing a gift with a probability of 80%. Also, 65% of all customers are female. Customers make purchasing decisions independently of each other.
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