The Binomial Probability Distribution and Related TopicsStatistical Experiments and Random VariablesRandom Variables and Their Probability DistributionsDiscrete Probability DistributionsProbability Distribution FeaturesMeans and Standard Deviations for Discrete Probability DistributionsLinear Functions of Random VariablesFinding µ and σ for Linear Functions of xIndependent Random VariablesCombining Random VariablesBinomial ExperimentsSlide 12Determining Binomial ProbabilitiesBinomial Probability FormulaUsing the Binomial TableSlide 16Binomial ProbabilitiesGraphing a Binomial DistributionMean and Standard Deviation of a Binomial DistributionCritical ThinkingQuota ProblemsThe Geometric DistributionSlide 23The Poisson DistributionSlide 25Finding Poisson Probabilities Using the TableUsing the Poisson TablePoisson Approximation to the BinomialThe Binomial Probability Distribution and Related TopicsChapter 5Understandable Statistics Ninth EditionBy Brase and Brase Prepared by Yixun ShiBloomsburg University of PennsylvaniaCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 2Statistical 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 | 3Random 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 | 4Discrete Probability Distributions1) 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 | 5Probability 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 µ for the mean and σ for the standard deviation.Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 6Means and Standard Deviations for Discrete Probability DistributionsCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 7Linear 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 | 8Finding µ and σ forLinear Functions of xCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 9Independent 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 | 10Combining Random VariablesCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 11Binomial Experiments1) 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 F = failureCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 12Binomial Experiments4) 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 | 13Determining Binomial Probabilities1) 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 | 14Binomial Probability FormulaCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 15Using the Binomial Table1) 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 | 16Recall for the sharpshooter example, n = 8, r = 6, p = 0.7So the probability she hits exactly 6 targets is 0.296, as expectedCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 17Binomial 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 | 18Graphing a Binomial DistributionCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 19Mean and Standard Deviation of a Binomial DistributionnpqnpCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5 | 20Critical 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 | 21Quota 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 | 22The 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 diskCopyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.