3.4. The Binomial Probability Distribution

Transcription

1 3.4. The Binomial Probability Distribution Objectives. Binomial experiment. Binomial random variable. Using binomial tables. Mean and variance of binomial distribution Four Conditions that determined a Binomial Experiment. 1. The experiment (composite) consists of a sequence of n more simple experiments called trials, where n is fixed in advance. 2. Each trial can result in one of the same two possible outcomes (dichotomous trials), denoted by success (S) and failure (F). 3. The trials are independent, so that the outcome of any particular trial does not influence the outcome on any other trial. P S is constant from trial to trial. 4. The probability of success PS will be denoted by p. Definition An experiment for which Conditions 1 4 are satisfied is called a binomial experiment. Examples of binomial experiments

2 Sampling without Replacement. Consider sampling without replacement from dichotomous population of size N Examples of Sample Spaces and Events. If the sample size (number of trials) n is at most 5%of the population size, For the the experiment following can random be analyzed experiments as though describe it were the exactly sample a space binomial and experiment. some events The Binomial RV and Distribution. Definition. The binomial random variable X associated with a binomial experiment consisting of n trials is defined as X the number of S s among the n trials Bin n, p Notation: Binomial random variable X has the following probability mass distribution x n 1 px, 0,1, 2,..., 1, p x P X x x n n n Notation for Binomial pmf Binomial pmf is denoted by b x; n, p Theorem Formula for b x; n, p b x; n, p n x nx p 1 p x 0,1, 2,..., n x 0 otherwise 2

3 Using Binomial Tables Formula for Binomial Cumulative Distribution Function Bx; n, p ; n p B x, P X x b y; n, p x 0,1, 2,..., n y0 Binomial random variable X itself is denoted by Binn, p x Example (Example 3.32 p. 118 textbook) Suppose that 20% of all copies of a particular textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test. Then X has a binomial distribution with n 15 and p 0.2. Solution. 1. The probability that at most 8 fail the test is 8 ;15,0.2 8;15, 0.2 P X b y B y0 which is the entry in the row and the column of the binomial table. From Appendix Table A.1, the probability is B8;15, The probability that exactly 8 fail is ;15, 0.2 7;15, 0.2 P X P X P X B B This answer is the difference between two consecutive entries in the column. 3. The probability that at least 8 fail is ;15, 0.2 P X P X B The probability that between 4 and 7, inclusive, fails is 7 P X 3! not P X 7 P X 4 P X! not 4 7 4, 5, 6, or 7 P X 7 P X 3 B7;15, 0. 2 B3;15, 0.2 P X P X

4 The Mean and Variance of X Binn, p. If X Binn, p, then, 1 X npq where q 1 p E X np V X np p npq and Example (Example 3.34 p. 120 textbook) If 75% of all purchases at a certain store are made with a credit card and X is the number among ten randomly selected purchases made with a credit card, then E X, V X, and P E X X E X. find Solution. X Bin 10, 0.75 E X np V X npq Interpretation of 7.5 If we perform a large number of independent binomial experiments, each with n 10 and p 0.75, then the average number of successes in the sequences will be close P6.13 X 8.87 P X 7 or 8 P E X X E X P X Bx from the table,10,0.75 4

5 3.6. The Poisson Probability Distribution. Objectives. Definition of Poisson distribution. Mean and variance of Poisson distribution. Poisson distribution as a limit. The Poisson process. Definition of Poisson distribution A discrete random variable X is said to have a Poisson distribution with 0 if the pmf of X is parameter x e ; p x x 0,1, 2, 3,... x! For Poisson distribution E X V X The Poisson Distribution as a Limit. In any binomial experiment in which n is large and p is small, bx; n, p px;, where np. As a rule of thumb, this approximation can safely be applied if n 50and np 5. The Poisson Distribution is called the Law of occurrence of rare events. General description of Poisson experiment. A Poisson experiment is a statistical experiment that has the following properties: The experiment results in outcomes that can be classified as successes or failures. The average number of successes (μ) that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero. Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc. 5

6 Example (example 3.40 p. 129 textbook) If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is.005 and errors are independent from page to page, what is the probability that one of its 400-page novels will contain exactly one page with errors? At most three pages with errors? Solution. S a page contains at least one error, F error-free page X the number of pages containing at least one error is 400, np Bin. 2 e P X 1 b1; p1; ! The value for the same probability computed by binomial distrubution b 1;400, So, the approximation is very good. at most three pages with errors 3,2 P P X p x 3 x0 e 2 x 2 x! 3 x For Bin 400, 0.005, P X

7 The Poisson Process. A very important application of the Poisson distribution arises in connection with the occurrence of events of some type over time. Events of interest might be visits to a particular website, pulses of some sort recorded by a counter, messages sent to a particular address, accidents in an industrial facility, or cosmic ray showers observed by astronomers at a particular observatory. We make the following assumptions about the way in which the events of interest occur: 1. There exists a parameter 0 such that for any short time interval of length t, the probability that exactly one event occurs is t t. 2. The probability of more than one event occurring during t is t. 3. The number of events occurring during the time interval t is independent of the number that occur prior to this time interval. Informally, assumption 1 says that for a short interval of time, the probability of a single event occurring is approximately proportional to the length of the time interval, where is the constant of proportionality. Denote by Pk t the probability that k events will be observed during any particular time interval of length t Description of Poisson process. t k e t Pk t so that the number of events during a time interval of k! length t is a Poisson rv with parameter t. The expected number of events during any such time interval is then t, so, the expected number during a unit interval of time is. The occurrence of events over time as described is called a Poisson process; the parameter specifies the rate for the process. 7

8 Example of Poisson process (example 3.42 p.131 textbook) Suppose pulses arrive at a counter at an average rate of six per minute, so that 6. To find the probability that in a.5-min interval at least one pulse is received, note that the number of pulses in such an interval has a Poisson distribution with t (.5 min is used because a is expressed as a rate per parameter minute). Then X the number of pulses received in the 30-sec interval, 0 e 3 3 P 1 X 1 P X ! 8

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