How are Geometric and Poisson probability distributions different from the binomial probability distribution? How are they the same?

Transcription

1 Probability and Statistics The Binomial Probability Distribution and Related Topics Chapter 5 Section 4 The Geometric and Poisson Probability Distributions Essential Question: How are Geometric and Poisson probability distributions different from the binomial probability distribution? How are they the same? Student Objectives: The student will determine the probability of an event using the geometric probability distribution. Poisson probability distribution. The student will use the Poisson probability distribution to approximate the probability of a binomial experiment. appropriate commands on their calculator. appropriate formula. appropriate chart. Terms: Geometric probability distribution Negative binomial distribution Poisson probability distribution

2 Equations: Geomeric Probability Distribution P( n) = p( 1 p) n 1 where n is the number of binomial trials on which the first success occurs (n = 1, 2, 3,...) and p is the probability of success on each trial. Notice: p is the same for each trial. Using some mathematics involving infinite series, it can be shown that the population mean and standard deviation of the geometric distribution are: µ= 1 p and σ = 1 p p Poisson Probability Distribution Let λ (Greek letter lambda) be the mean number of success over time, volume, area, and so forth. Let r, be the number of successes (r = 0, 1, 2, 3,...) in a corresponding interval of time, volume, area, and so forth. Then the probability of r successes in the interval is P( r) = e λ λ r where e is approximately equal to Using some mathematics involving infinite series, it can be shown that the population mean and standard deviation of the Poisson distribution are: µ = λ and σ = λ How to approximate Binomial Probabilities using Poisson Probabilities Suppose you have a binomial distribution with n = number of trials p = probability of succes in each trial, and r = the number of success If n 100 and np < 10, then r has a binomial distribution that is approximated by a Poisson distribution with λ = np. P r ( ) = e λ λ r Note: λ = np is the expected value of the binomial distribution.

3 Distributions Conditions and Settings Formulas Binomial Distributions Geometric Distributions Poisson Distributions 1. There a n independent each repeated under identical conditions. 2. Each trial has 2 outcomes: S = success and F = failure. 3. P(S) = p is the same for each trial, as is P(F) = q = 1 - p. 4. The random variable r represents the number of successes out of n trials. 0 r n 1. There a n independent each repeated under identical conditions. 2. Each trial has 2 outcomes: S = success and F = failure. 3. P(S) = p is the same for each trial, as is P(F) = q = 1 - p. 4. The random variable r represents the number of the trial on which the first success occurs. n = 1, 2, 3, Consider a random process that occurs over time, volume, area, or any other quantity that can (in theory) be divided into smaller and smaller intervals. 2. Identify success in the context of the interval (time, volume, area, ) you are studying. 3. Based on long-term experience, compute the mean or average number of success that occur over the interval (time, volume, area, ) you are studying. λ = mean number of successes over the designated interval 4. The random number r represents the number of successes that occur over the interval on which you perform the random process. r = 0, 1, 2, 3,... The probability of exactly r successes out of n trials is P( r) = n C r p r q n r For r, µ = np and σ = npq The probability for the first success occurs on the n th trial is P( n) = pq n 1 For n, µ = 1 p and σ = q p The probability of r success in the interval is ( ) = e λ λ r P r For r, µ = λ and σ = λ Poisson approximation to the binomial distribution 1. There a n independent each repeated under identical conditions. 2. Each trial has 2 outcomes: S = success and F = failure. 3. P(S) = p is the same for each trial, 4. In addition, n 100 and np The random variable r represents the number of successes out of n trials in a binomial experiment. λ = np is expected value of r. The probability of r successes on n trials is ( ) = e λ P r λ r

4 Negative Binomial Distribution Let k 1 be a fixed whole number. The probability that the kth success occurs on trial n is P( n) = n 1 C k 1 ( p) k ( q) n k where C = ( n 1)! n 1 k 1 ( k 1)! ( n k)! n = k, k +1, k + 2,... The expected value and standard deviation of the geometric distribution are: µ = k p and σ = kq p Note: if k = 1, the negative binomial distribution is called the geometric distribution. * Special Note: Fix the formula on page 238. * Graphing Calculator Skills: Geometric Probability Distribtion: a) Probaility of first success on a given trial: geometpdf( p, n). b) Probaility of first success up to and including a given trial: geometcdf( p, n). Poisson Probability Distribtion: a) Probaility of r successes on a given trial: poissonpdf( λ, r). b) Probaility of at least r success: poissoncdf( λ, r).

5 Sample Questions: 1. You are playing a carnival game where the winning coin is under one of 3 cups placed upside-down on a table. The cups are moved in a scrambled pattern so you cannot determine where the winning coin is at? What is the probability of winning the first time you play the game? The second time? The third time? How many games would you need to play to be 95% certain that you would win? Use the formula to answer the first three questions and the calculator to answer last answer. 2. What is the mean number of games required to win the game mentioned in question #1? What is the standard deviation?

6 3. It has been determined that about that 14 cars run a red-light in downtown Annville everyday. The police have decided to place a patrol car in the area for a 10-hour period. Round you value of λ to the nearest tenth and use the chart to determine the probability of 0, 1, and 2 cars running the red light. Use the formula and the exact value of λ to determine the probability of 5 cars running the red light. Use the calculator to determine the probability of 4 cars running the red light. Use the calculator to determine the probability that 7 or more cars will run the red light. 4. In a recent study of the high school graduates it has been determined that the probability that a graduate will be in serious trouble with the law within 5 years of the graduate is Is the Poisson distribution a good approximation of the binomial distribution? Explain. There are 144 students in the graduating class.

7 5. Use the formula for binomial probability to determine the probability that 4 students will get in serious trouble with the law within 5 years of graduation. Use your calculator to determine the probability that at least 10 students will get in serious trouble with the law. Repeat these calculations with the Poisson formulas. Are the answers similar? Homework: Pages Exercises: #1-17, odd (Calculator: 9d, 11b, 11c,11d(b), 11d(c), 13c, 15c, 15d, 17; Chart: 15b; Formula: 7, 9a, 9b, 9c, 13b) Exercises: #19-31, odd (Calculator: 19b4, 21c, 29 (use actual value of lambda); Chart: 19c, 23 (use the rounded value of lambda), 25 (use the rounded value of lambda); Formula: 19b1, 19b2, 19b3, 21b, 27, 31b) Exercises: #2-16, even (Calculator: 10d, 12, 14c, 14d, 16c; Chart: 8, 16b; Formula: 10a, 10b, 10c, 14b, 16d) Exercises: #18-32, even (Calculator: 18c, 18d2, 20, 24, 26c, 26d, 28; Chart: 18b, 22c, 22d; Formula: 18d1, 22b, 26b, 30a, 30b, 32a, 32b) Pages Exercises: #1-20