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09 Midterm Review-1

# X p x x e x μ 2 2 2 x p x e x x e x var x x 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 - = - = = 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
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.
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 - = = λt μ = λt σ 2 = = σ t

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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?
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. a) What is the probability that the next customer will purchase a gift?

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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 σ 26 Discrete...

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