Other discrete random variables There are many discrete random variables We see a few more in these lecture notes geometric negative binomial and hypergeometric among the univariate ones Discrete random variables could also be multivariate So we also study the multinomial distribution Discrete r v IV Juana Sanchez UCLA Dept of Statistics may ignore R code Stat 100A Introduction to Probability Sanchez 1 In the past lectures we studied the binomial which is really a family The shape of the pmf changes with n and p If we fix n at n 4 and change p from left to right p 0 1 to 0 5 top row to 0 9 to 0 99 bottom row Stat 100A Introduction to Probability Sanchez 3 Stat 100A Introduction to Probability Sanchez 2 If we keep p fixed at p 0 1 and change n from n 4 to n 20 to n 100 to n 1000 Stat 100A Introduction to Probability Sanchez 4 1 R code example New families X 0 1000 X from 0 to 1000 P dbinom X 1000 0 1 Binomial prob for X plot X P xlab BinomialX ylab Probability main p 0 1 n 1000 type h ylim c 0 1 Stat 100A Introduction to Probability Sanchez 5 The new random variables form also families Each distribution is really a family of distributions that changes when the parameter changes Stat 100A Introduction to Probability Sanchez 6 P Y y P x1 0 x2 0 xy 1 0 xy 1 1 p y 1 p I The Geometric Random Variable Let Xi 1 if the trial of an experiment results in a success and Xi 0 otherwise Let P Xi 1 p be constant across trials The trials are independent of each other What is the number of the trial in which the first success occurs Let s call this number Y Y is a new r v and its probability distribution is Stat 100A Introduction to Probability Sanchez 7 for y 1 2 countably infinite Example of variables with this distribution a The year that a dam is in service before it overflows b the number of automobiles going through a radar check before the first speeder is detected c the number of mistakes before getting it right d and many others Stat 100A Introduction to Probability Sanchez 8 2 The pmf for this problem is done in R as follows because R models number of trials before success not number of the trial where first success observed Example X 0 20 P dgeom X 1 3 plot X 1 P xlab Geometric X ylab Probability main p 1 3 type h ylim c 0 0 5 If one third of the persons donating blood at a clinic have O blood find the probability that the first O donor is the fourth donor of the day P Y 4 1 1 3 3 1 3 0 0987 Stat 100A Introduction to Probability Sanchez 9 10 Example Expected Value and Variance of a geometric random variable A recruiting firm finds that 30 percent of the applicants for a certain industrial job have received advanced training in computer programming Applicants are interviewed sequentially and selected at random from the pool a Find the probability that the first applicant who has received advanced training in programming is found on the fifth interview Y number of interview on which 1st applicant having advanced training is found P Y 5 1 0 3 4 0 3 0 072 E X 1 p Var X 1 p p2 How many times should you expect to roll a die to get a 6 E X 6 times 1 1 6 give or take the standard deviation i e give or take 5 4 times Stat 100A Introduction to Probability Sanchez Stat 100A Introduction to Probability Sanchez 11 Stat 100A Introduction to Probability Sanchez 12 3 II The negative binomial random variable If the parameter p changes the distribution changes The geometric distribution models the probabilistic behavior of the number of the trial on which the first success occurs in a sequence of independent Bernoulli trials But what if we were interested in the number of the trial for the second success or the third success or in general the rth success The distribution governing probabilistic behavior in these cases is called the negative binomial distribution Stat 100A Introduction to Probability Sanchez 13 Neg Binomial cont d 14 hence Let Y denote the number of the trial on which the rth success occurs in a sequence of independent Bernoulli trials P Y y P 1st y 1 trials contain r 1 successes and the yth trial is a success P 1st y 1 trials contain r 1 successes x P yth trial is a success The first probability statement is identical to the one that results in a binomial model and Stat 100A Introduction to Probability Sanchez Stat 100A Introduction to Probability Sanchez 15 y 1 r 1 y 1 r p 1 p y r p p 1 p y r P Y y r 1 r 1 y r r 1 Stat 100A Introduction to Probability Sanchez 16 4 The answer is Example Assume independent trials with 0 3 being the probability of finding a qualified candidate on any one trial Let Y denote the number of the trial on which the third qualified candidate is found Y can be reasonably assumed to have a negative binomial distribution 30 of the applicants for a certain position have received advanced training in computer programming Suppose that three jobs that require advanced programming training are open Find the probability that the third qualified applicant is found on the fifth interview if the applicants are interviewed sequentially and at random Stat 100A Introduction to Probability Sanchez 4 P Y 5 0 3 3 0 7 2 0 079 2 17 Expected Value and Variance of the negative binomial r E X p r 1 p Var X p2 In the example of the candidates for the jobs how many interviews should we expect to conduct to find the three experienced candidates 3 0 3 10 the expected value but we could need 4 88 more or less Interviews than that 4 830 is the standard deviation Stat 100A Introduction to Probability Sanchez 19 Stat 100A Introduction to Probability Sanchez 18 III The Hypergeometric Random Variable A sample of size n is to be chosen randomly without replacement from a box containing N balls of which k are white and N k are black If we let X denote the number of white balls selected then X is a hypergeometric random variably with parameters N k n The pmf of X is k N k x n x P X x N n x 1 k Stat 100A Introduction to Probability Sanchez 20 5 Example Example A warehouse contains ten printing machines four of which are defective A company randomly selects five of the machines for purchase What is the probability that all of the machines are Not defective Y of Non defectives N 10 n 5 k 6 number of non defective machines 6 4 5 …
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