The Poisson Distribution. The Poisson Distribution may be used as an approximation for a binomial distribution when n is large and p is small enough that.

Slides:

Advertisements

Similar presentations

AP Statistics 51 Days until the AP Exam

Advertisements

Think about the following random variables… The number of dandelions in a square metre of open ground The number of errors in a page of a typed manuscript.

Week11 Parameter, Statistic and Random Samples A parameter is a number that describes the population. It is a fixed number, but in practice we do not know.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 7 Probability.

Discrete Random Variables: The Poisson Distribution IBHLY1/Y2 - Santowski.

Flipping an unfair coin three times Consider the unfair coin with P(H) = 1/3 and P(T) = 2/3. If we flip this coin three times, the sample space S is the.

Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1.Midterms. 2.Hellmuth/Gold. 3.Poisson. 4.Continuous distributions.

Discrete Random Variables and Probability Distributions

1 1 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 5: Probability Distributions: Discrete Probability Distributions.

Engineering Probability and Statistics - SE-205 -Chap 4 By S. O. Duffuaa.

Section 2.6 Consider a random variable X = the number of occurrences in a “unit” interval. Let = E(X) = expected number of occurrences in a “unit” interval.

Statistics Lecture 11.

QBM117 Business Statistics

1 Random Variables and Discrete probability Distributions SESSION 2.

Chapter 5 Probability Theory. Rule 1 : The probability P(A) of any event A satisfies 0 < P(A) < 1 Rule 2 : If S is the sample space in a probability model,

Discrete Probability Distributions Binomial Distribution Poisson Distribution Hypergeometric Distribution.

4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,

Standard error of estimate & Confidence interval.

CA200 Quantitative Analysis for Business Decisions.

Standard Statistical Distributions Most elementary statistical books provide a survey of commonly used statistical distributions. The reason we study these.

Chapter 5 Some Discrete Probability Distributions.

Probability Distribution

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Basic Business Statistics.

Lecture 4 1 Discrete distributions Four important discrete distributions: 1.The Uniform distribution (discrete) 2.The Binomial distribution 3.The Hyper-geometric.

Poisson Random Variable Provides model for data that represent the number of occurrences of a specified event in a given unit of time X represents the.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 5.3.

Probabilistic and Statistical Techniques 1 Lecture 19 Eng. Ismail Zakaria El Daour 2010.

 A probability function is a function which assigns probabilities to the values of a random variable.  Individual probability values may be denoted by.

1 STAT 500 – Statistics for Managers STAT 500 Statistics for Managers.

1 Topic 3 - Discrete distributions Basics of discrete distributions Mean and variance of a discrete distribution Binomial distribution Poisson distribution.

 A probability function is a function which assigns probabilities to the values of a random variable.  Individual probability values may be denoted.

Onur DOĞAN.  asdaf. Suppose that a random number generator produces real numbers that are uniformly distributed between 0 and 100.  Determine the.

SESSION 35 & 36 Last Update 5 th May 2011 Discrete Probability Distributions.

1 1 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 5: Probability Distributions:

Statistics Lecture Notes Dr. Halil İbrahim CEBECİ Chapter 06 Random Variables and Discrete Probability Distributions.

Topic 3 - Discrete distributions Basics of discrete distributions - pages Mean and variance of a discrete distribution - pages ,

Some Common Discrete Random Variables. Binomial Random Variables.

Chapter 4-5 DeGroot & Schervish. Conditional Expectation/Mean Let X and Y be random variables such that the mean of Y exists and is finite. The conditional.

Normal Distribution * Numerous continuous variables have distribution closely resemble the normal distribution. * The normal distribution can be used to.

Review of Chapter

Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.

 A probability function is a function which assigns probabilities to the values of a random variable.  Individual probability values may be denoted.

Module 5: Discrete Distributions

Chapter 4. Random Variables - 3

THE POISSON DISTRIBUTION

Introduction A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. It may.

Part 2: Named Discrete Random Variables

Random Variables Lecture Lecturer : FATEN AL-HUSSAIN.

12.1 Discrete Probability Distributions (Poisson Distribution)

Lesson Poisson Probability Distribution. Objectives Understand when a probability experiment follows a Poisson process Compute probabilities of.

1 5.6 Poisson Distribution and the Poisson Process Some experiments result in counting the numbers of particular events occur in given times or on given.

Chapter 6 – Continuous Probability Distribution Introduction A probability distribution is obtained when probability values are assigned to all possible.

Chapter 3 Probability Distribution

4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,

The Poisson probability distribution

Math 4030 – 4a More Discrete Distributions

Poisson Random Variables

Discrete Random Variables

Engineering Probability and Statistics - SE-205 -Chap 4

Random variables (r.v.) Random variable

Lesson 98 – Poisson Distribution

Statistical Inference for Managers

Binomial Distributions

S2 Poisson Distribution.

Probability distributions

LESSON 11: THE POISSON DISTRIBUTION

STATISTICAL MODELS.

Some Discrete Probability Distributions Part 2

Elementary Statistics

Theorem 5.3: The mean and the variance of the hypergeometric distribution h(x;N,n,K) are:  = 2 = Example 5.10: In Example 5.9, find the expected value.

Presentation transcript:

The Poisson Distribution

The Poisson Distribution may be used as an approximation for a binomial distribution when n is large and p is small enough that np is of moderate size. The Poisson Distribution, with some parameter m, looks like:

Some examples of random variables that usually obey the Poisson Distribution: 1. The number of misprints on a page of a book. 2. The number of people in a community living to 100 years of age. 3. The number of wrong telephone numbers that are dialed in a day. 4. The number of customers entering a post office on a given day.

Mean and Variance of a Poisson Distribution Mean: E(X) = m Variance: Var(X) = m

Given a Poisson distribution and that m = 2, calculate: Answer: Answer: Answer: 0.594

Suppose that the number of typographical errors on a single page of a book has a Poisson distribution with a parameter of. Calculate the probability that there is at least one error on a page. Answer: Suppose that the probability that an item produced by a certain machine will be defective is 0.1. Find the probability that a sample of 10 items will contain at most 1 defective item. Solve this using Binomial distribution Solve this using Poisson distribution Answer: Answer:

A research project found that an important role in the bonding process of river otters is social grooming. After extensive observations, it is found that one group of river otters under study had a frequency of grooming of approximately 1.7 for each 10 minutes. Suppose you are observing river otters for 30 minutes. Let r = 0, 1, 2, … be a random variable that represents the number of times in a 30 minute interval that one otter grooms another. Find the probability that one otter will groom another four or more times during the 30-minute observation period. Answer:

Police department officials indicate that the average number of homicides per day in New York City is 5.4. Assuming a Poisson process, what is the probability that on any given day there will be at most four homicides? Answer: The director of the Social Services agency for one city receives an average of five complaints a day about lack of child support claims received. Assuming it follows a Poisson distribution, what is the probability that on any day the director will receive at least three lack of child support claims? Answer:

The random variable X has a Poisson distribution with mean 4. Calculate (a) (b) (c) Answers: (a) (b) (c) SPEC/HL1/17

The number of fish present in a given stretch of river has a Poisson distribution with mean 3.8. Find the probability that in this stretch there are at most three fish. Answer: The number of incorrectly dialed numbers made per hour by a telephone sales representative has a Poisson distribution. The probability that she dials all of the numbers correctly is Find the mean number of incorrectly dialed numbers per hour. Answer: 0.198

The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5. a) Find the probability that more than two accidents will occur on a given Monday. b) Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day. Answers: a) b) 0.246

The random variable X follows a Poisson distribution. Given that, find: a) the mean of the distribution; b) Answers: a) 2.99 b) 0.424