MC MA 116 - Chapter 5 The Binomial Probability Distribution and Related Topics (28 pages)

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Chapter 5 The Binomial Probability Distribution and Related Topics



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Chapter 5 The Binomial Probability Distribution and Related Topics

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Pages:
28
School:
Montgomery College
Course:
Ma 116 - Elements Of Statistics
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Chapter 5 The Binomial Probability Distribution and Related Topics Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Statistical Experiments and Random Variables Statistical Experiments any process by which measurements are obtained A quantitative variable x is a random variable if its value is determined by the outcome of a random experiment Random variables can be discrete or continuous Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 2 Random Variables and Their Probability Distributions Discrete random variables can take on only a countable or finite number of values Continuous random variables can take on countless values in an interval on the real line Probability distributions of random variables An assignment of probabilities to the specific values or a range of values for a random variable Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 3 Discrete Probability Distributions 1 Each value of the random variable has an assigned probability 2 The sum of all the assigned probabilities must equal 1 Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 4 Probability Distribution Features Since a probability distribution can be thought of as a relative frequency distribution for a very large n we can find the mean and the standard deviation When viewing the distribution in terms of the population use for the mean and for the standard deviation Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 5 Means and Standard Deviations for Discrete Probability Distributions Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 6 Linear Functions of Random Variables Let a and b be constants Let x be a random variable L a bx is a linear function of x Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 7 Finding and for Linear Functions of x Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 8 Independent Random Variables Let x1 and x2 be random variables Then the random variables are independent if any event of x1 is independent of any event of x2 Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 9 Combining Random Variables Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 10 Binomial Experiments 1 There are a fixed number of trials This is denoted by n 2 The n trials are independent and repeated under identical conditions 3 Each trial has two outcomes S success Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved F failure 5 11 Binomial Experiments 4 For each trial the probability of success p remains the same Thus the probability of failure is 1 p q 5 The central problem is to determine the probability of r successes out of n trials Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 12 Determining Binomial Probabilities 1 Use the Binomial Probability Formula 2 Use Table 3 of Appendix II 3 Use technology Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 13 Binomial Probability Formula Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 14 Using the Binomial Table 1 Locate the number of trials n 2 Locate the number of successes r 3 Follow that row to the right to the corresponding p column Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 15 Recall for the sharpshooter example n 8 r 6 p 0 7 So the probability she hits exactly 6 targets is 0 296 as expected Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 16 Binomial Probabilities At times we will need to calculate other probabilities P r k P r k P r k P r k Where k is a specified value less than or equal to the number of trials n Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 17 Graphing a Binomial Distribution Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 18 Mean and Standard Deviation of a Binomial Distribution np npq Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 19 Critical Thinking Unusual values For a binomial distribution it is unusual for the number of successes r to be more than 2 5 standard deviations from the mean This can be used as an indicator to determine whether a specified number of r out of n trials in a binomial experiment is unusual Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 20 Quota Problems We can use the binomial distribution table backwards to solve for a minimum number of trials In these cases we know r and p We use the table to find an n that satisfies our required probability Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 21 The Geometric Distribution Suppose that rather than repeat a fixed number of trials we repeat the experiment until the first success Examples Flip a coin until we observe the first head Roll a die until we observe the first 5 Randomly select DVDs off a production line until we find the first defective disk Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 22 Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 23 The Poisson Distribution This distribution is used to model the number of rare events that occur in a time interval volume area length etc Examples Number of auto accidents during a month Number of diseased trees in an acre Number of customers arriving at a bank Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 24 The Poisson Distribution Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 25 Finding Poisson Probabilities Using the Table We can use Table 4 of Appendix II instead of the formula 1 Find at the top of the table 2 Find r along the left margin of the table Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 26 Using the Poisson Table Recall 4 r 0 r 4 r 7 Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 27 Poisson Approximation to the Binomial Copyright Houghton Mifflin Harcourt Publishing Company All rights reserved 5 28


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