Binomial Probability Distribution in Java

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two good items from ve positions in total We conclude that P(X = 2) = 5 2 3 p q = 10p2 q3 2
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In an entirely similar way, we nd that P(X = 3) = and P(X = 4) = 5 4 p q = 5p4 q and P(X = 5) = p5 4 5 3 2 p q = 10p3 q2 3
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If we add all these probabilities together, we nd P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = q5 + 5pq4 + 10p2 q3 + 10p3 q2 + 5p4 q + p5 = (q + p) = 1
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since q + p = 1
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Note that the coef cients in the binomial expansion add up to 32, so all the points in the sample space have been used The occurrence of the binomial theorem here is one reason the probability distribution of X is called the binomial probability distribution The above situation can be generalized Suppose now that we have n independent trials, that X denotes the number of successes, and that P(S) = p and P(F ) = q = 1 p We see that P(X = x) = n x n x p q x for x = 0, 1, 2, , n
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This is the probability distribution function for the binomial random variable in general n x n x = (q+p)n = 1, We note that P(X = x) 0 and n P(X = x) = n x=0 x=0 x p q so the properties of a discrete probability distribution function are satis ed Graphs of binomial distributions are interesting We show some here where we have chosen p = 03 for various values of n (Figures 53, 54, and 55) The graphs indicate that as n increases, the probability distributions become more bell shaped and strongly resemble what we will call, in 8, a continuous normal curve This is in fact the case, although this fact will not be pursued here One reason for not pursuing this is that exact calculations involving the
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035 03 025 02 015 01 005 0
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Random Variables and Discrete Probability Distributions
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Binomial distribution, n = 5,
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Binomial distribution, n = 15,
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015 0125 01 0075 005 0025 5 10 15 X 20 25 30
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Binomial distribution n = 30,
binomial distribution are possible using a statistical calculator or a computer algebra system, and so we do not need to approximate these probabilities with a normal curve, which we study in the chapter on continuous distributions Here are some examples
EXAMPLE 51
A Production Line
A production line has been producing good parts with probability 085 A sample of 20 parts is taken, and it is found that 4 of these are defective Assuming a binomial model, is this a cause for concern
Binomial Probability Distribution
Let X denote the number of good parts in a sample of 20 Assuming that the probability a good part is 085, we nd the probability the sample has at most 16 good parts is
P(X 16) =
20 (085)x (015)20 x = 0352275 x
So this event is not unusual and one would probably conclude that the production line was behaving normally and that although the percentage of defective parts has increased, the sample is not a cause for concern
EXAMPLE 52
A Political Survey
A sample survey of 100 voters is taken where actually 45% of the voters favor a certain candidate What is the probability that the sample will contain between 40% and 60% of voters who favor the candidate We presume that a binomial model is appropriate Note that the sample proportion of voters, say ps , can be expressed in terms of the number of voters, say X, who favor the candidate In fact, ps = X/100, so P(040 ps 060) = P 040 X 060 100
= P(40 X 60)
x=40