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5/13/12 Negative Binomial Distribution Contact us | Tell a friend | Search site Stat Trek Teach yourself statistics Home Tutorials AP Statistics Stat Tables Overview Stat Tools AP statistics Calculators Books Statistics and probability Help Matrix algebra Share on Google+ 6 Statistics Tutorial Negative Binomial Distribution Descriptive Statistics In this lesson, we cover the negative binomial distribution and the geometric distribution. As we will ► Quantitative measures see, the geometric distribution is a special case of the negative binomial distribution. ► Charts and graphs ► Tabular displays Probability Negative Binomial Experiment A negative binomial experiment is a statistical experiment that has the following properties: ► Probability basics The experiment consists of x repeated trials. ► Probability laws Each trial can result in just two possible outcomes. We call one of these outcomes a success ► Random variables and the other, a failure. ► Sampling theory The probability of success, denoted by P, is the same on every trial. Distributions The trials are independent; that is, the outcome on one trial does not affect the outcome on other ► Distribution basics trials. ▼ Discrete The experiment continues until r successes are observed, where r is specified in advance. Binomial distribution Negative binomial Hypergeometric Multinomial Poisson ► Continuous Estimation Consider the following statistical experiment. You flip a coin repeatedly and count the number of times List Price: $99.99 Buy Used: $67.99 Buy New: $76.42 Approved for AP Statistics and Calculus Statistics in Plain English, Third Edition Timothy C. Urdan the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because: The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads. Each trial can result in just two possible outcomes heads or tails. ► Estimation theory The probability of success is constant 0.5 on every trial. ► Proportions The trials are independent; that is, getting heads on one trial does not affect whether we get ► Mean scores heads on other trials. Hypothesis Testing The experiment continues until a fixed number of successes have occurred; in this case, 5 ► Foundations of testing heads. ► Mean scores Casio USB Graphing Calculator Model FX9860GII List Price: $39.95 Buy Used: $21.99 Buy New: $33.93 AP Statistics: NEW 3rd Edition (Advanced Placement (AP) Test Preparation) Robin LevineWissing, David Thiel Notation ► Proportions ► Power The following notation is helpful, when we talk about negative binomial probability. ► Chisquare tests x: The number of trials required to produce r successes in a negative binomial experiment. Survey Sampling r: The number of successes in the negative binomial experiment. ► Sampling methods P: The probability of success on an individual trial. ► Simple random samples Q: The probability of failure on an individual trial. (This is equal to 1 P.) ► Stratified samples ► Cluster samples ► Sample planning More Applied Statistics b*(x; r, P): Negative binomial probability the probability that an xtrial negative binomial Principles of Statistics (Dover Books on Mathematics) M.G. Bulmer experiment results in the rth success on the xth trial, when the probability of success on an individual trial is P. nC r: The number of combinations of n things, taken r at a time. ► Linear regression ► Regression tests List Price: $19.95 Buy Used: $15.00 Buy New: $149.99 Negative Binomial Distribution stattrek.com/probability-distributions/negative-binomial.aspx?Tutorial=Stat List Price: $16.95 Buy Used: $3.39 1/4 5/13/12 ► Experiments Appendices ■ Notation ■ Statistics Formulas Negative Binomial Distribution Buy New: $9.54 A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal distribution. Excel 2007 Data Analysis For Dummies Stephen L. Nelson Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a negative binomial experiment. The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below. Number of coin flips Probability 2 0.25 3 0.25 4 0.1875 5 0.125 6 0.078125 7 or more 0.109375 List Price: $24.99 Buy Used: $10.99 Buy New: $16.49 Statistics Hacks: Tips & Tools for Measuring the World and Beating the Odds Bruce Frey Negative Binomial Probability The negative binomial probability refers to the probability that a negative binomial experiment results in r 1 successes after trial x 1 and r successes after trial x. For example, in the above table, we see List Price: $29.99 Buy Used: $1.75 Buy New: $19.89 that the negative binomial probability of getting the second head on the sixth flip of the coin is 0.078125. Given x, r, and P, we can compute the negative binomial probability based on the following formula: Statistics (Cliffs Quick Review) David H. Voelker, Peter Z. Orton, Scott Adams Negative Binomial Formula. Suppose a negative binomial experiment consists of x trials and results in r successes. If the probability of success on an individual trial is P, then the negative binomial probability is: b*(x; r, P) = x1Cr1 * Pr * (1 P)x r The Mean of the Negative Binomial Distribution If we define the mean of the negative binomial distribution as the average number of trials required to produce r successes, then the mean is equal to: List Price: $9.99 Buy Used: $0.01 Buy New: $1.98 Practical Statistics Simply Explained (Dover Books Explaining Science) Russell Langley μ = r / P where μ is the mean number of trials, r is the number of successes, and P is the probability of a success on any given trial. List Price: $16.95 Buy Used: $0.01 Buy New: $10.10 Alternative Views of the Negative Binomial Distribution As if statistics weren't challenging enough, the above definition is not the only definition for the negative binomial distribution. Two common alternative definitions are: Probability For Dummies Deborah Rumsey The negative binomial random variable is R, the number of successes before the binomial experiment results in k failures. The mean of R is: μR = kP/Q The negative binomial random variable is K, the number of failures before the binomial experiment results in r successes. The mean of K is: List Price: $19.99 Buy Used: $4.87 Buy New: $11.90 μK = rQ/P stattrek.com/probability-distributions/negative-binomial.aspx?Tutorial=Stat 2/4 5/13/12 Negative Binomial Distribution The moral: If someone talks about a negative binomial distribution, find out how they are defining the negative binomial random variable. On this web site, when we refer to the negative binomial distribution, we are talking about the What is a pvalue anyway? 34 Stories to Help You Actually Understand Statistics Andrew J. Vickers definition presented earlier. That is, we are defining the negative binomial random variable as X, the total number of trials required for the binomial experiment to produce r successes. List Price: $33.33 Buy Used: $24.00 Buy New: $27.77 Geometric Distribution The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes (r) is equal to 1. Texas Instruments TI86 Graphing Calculator An example of a geometric distribution would be tossing a coin until it lands on heads. We might ask: What is the probability that the first head occurs on the third flip? That probability is referred to as a geometric probability and is denoted by g(x; P). The formula for geometric probability is given below. Geometric Probability Formula. Suppose a negative binomial experiment consists of x trials and results in one success. If the probability of success on an individual trial is P, then the geometric probability is: g(x; P) = P * Qx 1 List Price: $150.00 Buy Used: $2.79 Buy New: $189.85 Approved for AP Statistics and Calculus Sample Problems The problems below show how to apply your newfound knowledge of the negative binomial distribution (see Example 1) and the geometric distribution (see Example 2). Negative Binomial Calculator As you may have noticed, the negative binomial formula requires some potentially time consuming computations. The Negative Binomial Calculator can do this work for you quickly, easily, and errorfree. Use the Negative Binomial Calculator to compute negative binomial probabilities and geometric probabilities. The calculator is free. It can be found under the Stat Tables tab, which appears in the header of every Stat Trek web page. Negative Binomial Calculator Example 1 Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot? Solution: This is an example of a negative binomial experiment. The probability of success (P) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 3. To solve this problem, we enter these values into the negative binomial formula. b*(x; r, P) = x1Cr1 * Pr * Qx r b*(5; 3, 0.7) = 4C2 * 0.73 * 0.32 b*(5; 3, 0.7) = 6 * 0.343 * 0.09 = 0.18522 Thus, the probability that Bob will make his third successful free throw on his fifth shot is 0.18522. Example 2 stattrek.com/probability-distributions/negative-binomial.aspx?Tutorial=Stat 3/4 5/13/12 Negative Binomial Distribution Let's reconsider the above problem from Example 1. This time, we'll ask a slightly different question: What is the probability that Bob makes his first free throw on his fifth shot? Solution: This is an example of a geometric distribution, which is a special case of a negative binomial distribution. Therefore, this problem can be solved using the negative binomial formula or the geometric formula. We demonstrate each approach below, beginning with the negative binomial formula. The probability of success (P) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 1. We enter these values into the negative binomial formula. b*(x; r, P) = x1Cr1 * Pr * Qx r b*(5; 1, 0.7) = 4C0 * 0.71 * 0.34 b*(5; 3, 0.7) = 0.00567 Now, we demonstate a solution based on the geometric formula. g(x; P) = P * Qx 1 g(5; 0.7) = 0.7 * 0.34 = 0.00567 Notice that each approach yields the same answer. < Previous lesson Next lesson > About Us Contact Us Privacy Terms of Use Site Map Tell a Friend Resources Advertising The contents of this webpage are copyright © 2012 StatTrek.com. 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