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Chapter 4. Discrete Probability Distributions Section 4.6: Negative Binomial Distribution Jiaping Wang Department of Mathematical Science 02/25/2013, Monday The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Outline Probability Function Mean and Variance Examples The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 1. Probability Function The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL In the last section, the geometric distribution models the probability of the number of failures prior to the first success in a sequence of independent Bernoulli trials. What if we were interested in the number of failures prior to the second success, or the third success or (in general) the r-th success? Let X denote the number of failures prior to the r-th success, p denotes the common probability. P(X=x) = p(x) =P(The 1st (x+r-1) trials contain (r-1) successes and (x+r)th trial is a success) = P(The 1st (x+r-1) trials contain (r-1) successes)P((x+r)th trial is a success) − 1 1 − = +−1 1 − = +−1 −1 −1 For x=0, 1, … The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Probability Function The negative binomial distribution function: , x= 0, 1, 2, …., q=1-p P(X=x)=p(x)= +−1 −1 If r=1, then the negative binomial distribution becomes the geometric distribution. Example 4.19: As in Example 4.15, 20% of the applicants for a certain sales position Are fluent in English and Spanish. Suppose that four jobs requiring fluency in English And Spanish are open. Find the probability that two unqualified applicants are Interviewed before finding the fourth qualified applicant, if the applicants are interviewed sequentially and at random. Answer: so r=4, x=2, p=0.2, p(X=2)= 2+4−1 4−1 0.24(1-0.2)2=10*(0.2)4(0.8)2=0.01 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 2. Mean and Variance The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Mean and Variance Let W1 be the number of failures prior to the 1st success, W2 be the number of failures between 1st and 2nd success, so for W3, …, then X=∑Wi, and Wi follows the geometric distribution and independently, So E(X)=E(∑Wi)= ∑q/p=rq/p and V(X)= ∑V(Wi)=rq/p2. In summary, = , = 2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example Suppose we are at a rifle range with an old gun that misfires 5 out of 6 times. Dene “success" as the event the gun fires and let X be the number of failures before the third success. Then X is a negative binomial random variable with parameters (3, 1/6 ). Find E(X) and Var(X). Answer: E(X)=rq/p=3*5/6*6=15, V(X)=rq/p2=3*5/6*36=90. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL An alternative way, if let n=x+r, then the probability function becomes −1 = = = 1 − − , = , + 1, + 2, … −1 Which can be used to model the count data, such as the number of accidents in a year, the number of trees in a plot, or the number of insects on a plant. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 4.21 Barnacle often attach to hulls of ships. Their presence speeds corrosion and increases drag resistance, leading to reduced speed and maneuverability. Let X denote the number of barnacles on a randomly selected square meter of a ship hull. For a particular shipyard, the mean and variance of X are 0.5 and 0.625, respectively. Find the probability that at least one barnacle will be on a randomly selected meter of a ship hull. Answer: E(X)=r(1-p)/p=0.5, V(X)=r(1-p)/p2=0.625, so we have V(X)=E(X)/p=0.5/p=0.625 p=0.5/0.625=0.8, so P(X ≥ 1)=1-P(X=0)=10.8*0.8=0.36. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example A research scientist is inoculating rabbits, one at a time, with a disease until he finds two rabbits which develop the disease. If the probability of contracting the disease 1/6. What is the probability that eight rabbits are needed? Let X be the number of rabbits needed until the first rabbit to contract the disease. Then X follows a negative binomial distribution with r = 2; x = 6; and p = 1/6. Thus, P(X=6)=C(2+6-1,6)(1/6)2(5/6)6 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example Suppose that 3% of computer chips produced by a certain machine are defective. The chips are put into packages of 20 chips for distribution to retailers. (a) What is the probability that a randomly selected package of chips will contain at least 2 defective chips? (b) What is the probability that the ten-th pack selected is the third to contain at least two defective chips? (a) n=20, p=3%, so P(X≥2)=1-P(X=0)-P(X=1)=120 3% 1 97% 20 1 (b) r=3, x=10, p=P(X ≥2), so P(X=10)= 3+10−1 10 20 0 3% ≥2 0 97% 3[1 20 − − ( ≥ 2)]10 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL HW#6 Page 150: 4.66, 4.67 Page 152: 4.84. Additional Problem: 1. Find the expected value and the variance of the number of times one must throw a die until the outcome 1 has occurred 4 times. 2. If the probability is 0.40 that a child exposed to a certain contagious disease will catch it, what is the probability that the tenth child exposed to the disease will be the third to catch it? Due Wed., 03/06/2013 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL