Approximating distributions

Slides:

Advertisements

Similar presentations

Special random variables Chapter 5 Some discrete or continuous probability distributions.

Advertisements

Statistics S2 Year 13 Mathematics. 17/04/2015 Unit 1 – The Normal Distribution The normal distribution is one of the most important distributions in statistics.

Probability Distribution

Normal Approximations to Binomial Distributions Larson/Farber 4th ed1.

Normal Approximation of the Binomial Distribution.

Normal Probability Distributions

Modeling Process Quality

Chapter 6 The Normal Distribution

Statistics Lecture 11.

Chapter Five Continuous Random Variables McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

Some Continuous Probability Distributions

The Poisson Probability Distribution The Poisson probability distribution provides a good model for the probability distribution of the number of “rare.

JMB Ch6 Lecture 3 revised 2 EGR 252 Fall 2011 Slide 1 Continuous Probability Distributions Many continuous probability distributions, including: Uniform.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 8 Continuous.

Probability theory 2 Tron Anders Moger September 13th 2006.

Normal Approximation Of The Binomial Distribution:

Section 5.5 Normal Approximations to Binomial Distributions Larson/Farber 4th ed.

Continuous Random Variables

Distributions Normal distribution Binomial distribution Poisson distribution Chi-square distribution Frequency distribution

Binomial Distributions Calculating the Probability of Success.

4.4 Normal Approximation to Binomial Distributions

5.5 Normal Approximations to Binomial Distributions Key Concepts: –Binomial Distributions (Review) –Approximating a Binomial Probability –Correction for.

Bluman, Chapter 61. Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients,

Using Normal Distribution to Approximate a Discrete Distribution.

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.

Copyright ©2011 Nelson Education Limited The Normal Probability Distribution CHAPTER 6.

Lecture 7.  To understand what a Normal Distribution is  To know how to use the Normal Distribution table  To compute probabilities of events by using.

Introduction Discrete random variables take on only a finite or countable number of values. Three discrete probability distributions serve as models for.

Geometric Distribution

MTH3003 PJJ SEM I 2015/2016.  ASSIGNMENT :25% Assignment 1 (10%) Assignment 2 (15%)  Mid exam :30% Part A (Objective) Part B (Subjective)  Final Exam:

Biostat. 200 Review slides Week 1-3. Recap: Probability.

Chapter 6 Some Continuous Probability Distributions.

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 5 Discrete Random Variables.

McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 5 Discrete Random Variables.

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 7 Section 5 – Slide 1 of 21 Chapter 7 Section 5 The Normal Approximation to the Binomial.

Chapter 12 Probability. Chapter 12 The probability of an occurrence is written as P(A) and is equal to.

Normal Approximations to Binomial Distributions. 

Continuous Random Variables Continuous random variables can assume the infinitely many values corresponding to real numbers. Examples: lengths, masses.

MATH 4030 – 4B CONTINUOUS RANDOM VARIABLES Density Function PDF and CDF Mean and Variance Uniform Distribution Normal Distribution.

Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 5-5 Poisson Probability Distributions.

Physics 270. o Experiment with one die o m – frequency of a given result (say rolling a 4) o m/n – relative frequency of this result o All throws are.

Normal approximation of Binomial probabilities. Recall binomial experiment:  Identical trials  Two outcomes: success and failure  Probability for success.

381 More on Continuous Probability Distributions QSCI 381 – Lecture 20.

THE NORMAL APPROXIMATION TO THE BINOMIAL. Under certain conditions the Normal distribution can be used as an approximation to the Binomial, thus reducing.

Normal Approximations to Binomial Distributions.  For a binomial distribution:  n = the number of independent trials  p = the probability of success.

Chapter 4. Random Variables - 3

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Normal Probability Distributions 5.

Lesson The Normal Approximation to the Binomial Probability Distribution.

Statistics -Continuous probability distribution 2013/11/18.

12.SPECIAL PROBABILITY DISTRIBUTIONS

Normal distribution 1. Learn about the properties of a normal distribution 2. Solve problems using tables of the normal distribution or your calculator.

Normal Approximation to the Binomial Distribution.

MATB344 Applied Statistics

The Recursive Property of the Binomial Distribution

Chapter 5 Normal Probability Distributions.

Discrete Probability Distributions

MTH 161: Introduction To Statistics

Example of Poisson Distribution-Wars by Year

Continuous Random Variables

Normal Distribution.

Do now: • Which diagram on the right illustrates which distribution?

Normal Density Curve. Normal Density Curve 68 % 95 % 99.7 %

In a recent year, the American Cancer said that the five-year survival rate for new cases of stage 1 kidney cancer is 95%. You randomly select 12 men who.

Continuous Random Variable Normal Distribution

Exam 2 - Review Chapters

Chapter 5 Section 5-5.

If the question asks: “Find the probability if...”

Introduction to Probability and Statistics

Elementary Statistics

Chapter 5 Normal Probability Distributions.

Presentation transcript:

Approximating distributions Binomial distribution Approximating distributions Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial to Normal Binomial distribution Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating the binomial distribution with a normal distribution has discrete data and looks like Normal has continuous data and looks like f 0.15 0.1 0.05 x 2 4 6 8 10 12 Approximating distributions: Nannestad

Approximating distributions: Nannestad Why bother? Tables are not readily available for large values of n. When n is large, and p is not too close to either 0 or 1, then the binomial values are reasonably symmetrical. In these conditions, the normal curve closely fits the binomial values. Approximating distributions: Nannestad

Approximating distributions: Nannestad How do we approximate a binomial distribution with a normal distribution? The mean of a binomial distribution with n terms and probability p is np, and the variance is np(1-p). Use these values as the mean and variance for a normal distribution. Approximating distributions: Nannestad

When is the normal approximation “good enough”? Look at the following examples. When is the normal curve “close enough” to the binomial distribution? Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.10 Normal: mean = 1.50, VAR = 1.35 f Binomial: n = 15, p = 0.10 0.3 Normal: mean = 1.50, VAR = 1.35 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0. 15 Normal: mean = 2.25, VAR = 1.91 f Binomial: n = 15, p = 0.15 0.3 Normal: mean = 2.25, VAR = 1.91 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.20 Normal: mean = 3.00, VAR = 2.40 f Binomial: n = 15, p = 0.20 0.3 Normal: mean = 3.00, VAR = 2.40 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.25 Normal: mean = 3.75, VAR = 2.81 f Binomial: n = 15, p = 0.25 0.3 Normal: mean = 3.75, VAR = 2.81 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.30 Normal: mean = 4.50, VAR = 3.15 f Binomial: n = 15, p = 0.30 0.3 Normal: mean = 4.50, VAR = 3.15 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.35 Normal: mean = 5.25, VAR = 3.41 f Binomial: n = 15, p = 0.35 0.3 Normal: mean = 5.25, VAR = 3.41 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.40 Normal: mean = 6.00, VAR = 3.60 f Binomial: n = 15, p = 0.40 0.3 Normal: mean = 6.00, VAR = 3.60 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.45 Normal: mean = 6.75, VAR = 3.71 f Binomial: n = 15, p = 0.45 0.3 Normal: mean = 6.75, VAR = 3.71 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.50 Normal: mean = 7.50, VAR = 3.75 f Binomial: n = 15, p = 0.50 0.3 Normal: mean = 7.50, VAR = 3.75 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.55 Normal: mean = 8.25, VAR = 3.71 f Binomial: n = 15, p = 0.55 0.3 Normal: mean = 8.25, VAR = 3.71 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.60 Normal: mean = 9.00, VAR = 3.60 f Binomial: n = 15, p = 0.60 0.3 Normal: mean = 9.00, VAR = 3.60 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.65 Normal: mean = 9.75, VAR = 3.41 f Binomial: n = 15, p = 0.65 0.3 Normal: mean = 9.75, VAR = 3.41 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.70 Normal: mean = 10.5, VAR = 3.15 f Binomial: n = 15, p = 0.70 0.3 Normal: mean = 10.5, VAR = 3.15 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.75 Normal: mean = 11.3, VAR = 2.81 f Binomial: n = 15, p = 0.75 0.3 Normal: mean = 11.3, VAR = 2.81 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.80 Normal: mean = 12.0, VAR = 2.40 f Binomial: n = 15, p = 0.80 0.3 Normal: mean = 12.0, VAR = 2.40 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.85 Normal: mean = 12.8, VAR = 1.91 f Binomial: n = 15, p = 0.85 0.3 Normal: mean = 12.8, VAR = 1.91 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.90 Normal: mean = 13.5, VAR = 1.35 f Binomial: n = 15, p = 0.90 0.3 Normal: mean = 13.5, VAR = 1.35 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.95 Normal: mean = 14.3, VAR = 0.71 f Binomial: n = 15, p = 0.95 0.3 Normal: mean = 14.3, VAR = 0.71 0.2 0.1 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad General guidelines We can approximate the binomial distribution with n terms and probability p with the corresponding normal distribution when both np5, and n(1-p)5. Approximating distributions: Nannestad

Approximating distributions: Nannestad For the example with n = 15 and p changes by 0.05 each time, this would require p values 0.35  p  0.65 Approximating distributions: Nannestad

What happens to the values when we change from discrete data to continuous data? In a binomial distribution we can have a probability for a single value. Example: For binomial n = 15, p = 0.40, P(X=4) = 0.1268. However, it is not possible to have proability for a single value in normal distribution. Example: Normal: mean = 6.00, VAR = 3.60, P(X=4) cannot exist. Why? Approximating distributions: Nannestad

The probability is calculated from the area under the normal curve. For the area to be other than zero we need to find the probability between two values Approximating distributions: Nannestad

Continuity correction When approximating a discrete distribution with one that is continuous we must apply a continuity correction. Original binomial distribution n = 30, p = 0.7 The value for P(X = 20) in the binomial distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial to Normal Binomial distribution Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial to Poisson Binomial distribution Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad When the binomial value for n is large and p is very small the event is considered “rare” and we can approximate values with the Poisson distribution. Approximating distributions: Nannestad

Approximating the binomial distribution with a poisson distribution has discrete data and looks like Poisson has disctrete data and looks like f 0.3 0.2 0.1 x Approximating distributions: Nannestad 2 4 6 8 10

Approximating distributions: Nannestad When? Why? How? When the value of p is too small for the tables, the probability of the event occuring is becoming “rare”. (np < 5 is a fair guide.) Binomial tables do not give the required values. Using np = , the binomial distribution values are approximated by the Poisson distribution. Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.005 Poisson:  = 0.075 f 1 0.8 Binomial: n = 15, p = 0.005 Poisson: lamda = 0.075 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.010 Poisson:  = 0.150 f 1 0.8 Binomial: n = 15, p = 0.010 Poisson: lamda = 0.150 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.015 Poisson:  = 0.225 f 1 0.8 Binomial: n = 15, p = 0.015 Poisson: lamda = 0.225 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.020 Poisson:  = 0.300 f 1 0.8 Binomial: n = 15, p = 0.020 Poisson: lamda = 0.300 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.025 Poisson:  = 0.375 f 1 0.8 Binomial: n = 15, p = 0.025 Poisson: lamda = 0.375 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.030 Poisson:  = 0.450 f 1 0.8 Binomial: n = 15, p = 0.030 Poisson: lamda = 0.450 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.035 Poisson:  = 0.525 f 1 0.8 Binomial: n = 15, p = 0.035 Poisson: lamda = 0.525 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.040 Poisson:  = 0.600 f 1 0.8 Binomial: n = 15, p = 0.040  Poisson: lamda = 0.600 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.045 Poisson:  = 0.675 f 1 0.8 Binomial: n = 15, p = 0.045 Poisson: lamda = 0.675 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.050 Poisson:  = 0.750 f 1 0.8 Binomial: n = 15, p = 0.050 Poisson: lamda = 0.750 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.055 Poisson:  = 0.825 f 1 0.8 Binomial: n = 15, p = 0.055 Poisson: lamda = 0.825 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.060 Poisson:  = 0.900 f 1 0.8 Binomial: n = 15, p = 0.060 Poisson: lamda = 0.900 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.065 Poisson:  = 0.975 f 1 0.8 Binomial: n = 15, p = 0.065 Poisson: lamda = 0.975 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.070 Poisson:  = 1.050 f 1 0.8 Binomial: n = 15, p = 0.070 Poisson: lamda = 1.050 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.075 Poisson:  = 1.125 f 1 0.8 Binomial: n = 15, p = 0.075 Poisson: lamda = 1.125 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.080 Poisson:  = 1.200 f 1 0.8 Binomial: n = 15, p = 0.080 Poisson: lamda = 1.200 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.085 Poisson:  = 1.275 f 1 0.8 Binomial: n = 15, p = 0.085 Poisson: lamda = 1.275 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.090 Poisson:  = 1.350 f 1 0.8 Binomial: n = 15, p = 0.090 Poisson: lamda = 1.350 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.095 Poisson:  = 1.425 f 1 0.8 Binomial: n = 15, p = 0.095 Poisson: lamda = 1.425 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.100 Poisson:  = 1.500 f 1 0.8 Binomial: n = 15, p = 0.100 Poisson: lamda = 1.500 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.15 Poisson:  = 2.25 f 1 0.8 Binomial: n = 15, p = 0.15 Poisson: lamda = 2.25 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.20 Poisson:  = 3.00 f 1 0.8 Binomial: n = 15, p = 0.20 Poisson: lamda = 3.00 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.25 Poisson:  = 3.75 f 1 0.8 Binomial: n = 15, p = 0.25 Poisson: lamda = 3.75 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.30 Poisson:  = 4.50 f 1 0.8 Binomial: n = 15, p = 0.30 Poisson: lamda = 4.50 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.35 Poisson:  = 5.25 f 1 0.8 Binomial: n = 15, p = 0.35 Poisson: lamda = 5.25 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.40 Poisson:  = 6.00 f 1 0.8 Binomial: n = 15, p = 0.40 Poisson: lamda = 6.00 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.45 Poisson:  = 6.75 f 1 0.8 Binomial: n = 15, p = 0.45 Poisson: lamda = 6.75 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0. 50 Poisson:  = 7.50 f 1 0.8 Binomial: n = 15, p = 0.50 Poisson: lamda = 7.50 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.55 Poisson:  = 8.25 f 1 0.8 Binomial: n = 15, p = 0.55 Poisson: lamda = 8.25 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.60 Poisson:  = 9.00 f 1 0.8 Binomial: n = 15, p = 0.60 Poisson: lamda = 9.00 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial: n = 15, p = 0.65 Poisson:  = 9.75 f 1 0.8 Binomial: n = 15, p = 0.65 Poisson: lamda = 9.75 0.6 0.4 0.2 x 2 4 6 8 10 Approximating distributions: Nannestad

Approximating distributions: Nannestad Binomial to Poisson Binomial distribution Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson to Normal Binomial distribution Normal distribution Poisson distribution Approximating distributions: Nannestad

Approximating the Poisson distribution with a normal distribution has discrete data and looks like Normal has continuous data and looks like 0.2 f 0.2 f 0.15 0.15 0.1 0.1 0.05 0.05 x x 5 10 15 20 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Why? When n is large and  >20 the poisson distribution is closely approximated by the normal distribution. This is a reason tables do not have  values bigger than 20. Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 9.00 Normal: mean = 12.8, VAR = 1.91 f . 1 5 . 1 . 5 x 1 2 Approximating distributions: Nannestad

Continuity correction Approximating distributions: Nannestad

Approximating distributions: Nannestad How? Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 10 Normal: mean = 10, VAR = 10 . 2 f . 1 5 . 1 . 5 x 5 1 1 5 2 Approximating distributions: Nannestad

Approximating distributions: Nannestad Drawing a normal curve with mean = variance = 9, and drawing the Poisson and normal distributions together allows comparison. Approximating distributions: Nannestad

Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 9 Normal: mean = 9, VAR = 9 0.2 f 0.15 0.1 0.05 x 5 10 15 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad For this value of  = 9 the approximation of a normal curve (mean = variance = 9) is not close, so in this case we would use values from the Poisson table. Approximating distributions: Nannestad

Approximating distributions: Nannestad

Approximating distributions: Nannestad Why? When n is large and  >20 the Poisson distribution is closely approximated by values from the normal distribution. This is a reason tables do not have  values bigger than 20. Approximating distributions: Nannestad

Approximating distributions: Nannestad Consider how close the normal curve is to the Poisson values as  increases. When n is large and  >20 the Poisson distribution is closely approximated by the normal distribution. Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 5 Normal: mean = 5, VAR = 5 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 10 Normal: mean = 10, VAR = 10 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 15 Normal: mean = 15, VAR = 15 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 16 Normal: mean = 16, VAR = 16 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 17 Normal: mean = 17, VAR = 17 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 18 Normal: mean = 18, VAR = 18 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 19 Normal: mean = 19, VAR = 19 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 20 Normal: mean = 20, VAR = 20 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 21 Normal: mean = 21, VAR = 21 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 22 Normal: mean = 22, VAR = 22 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 23 Normal: mean = 23, VAR = 23 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 24 Normal: mean = 24, VAR = 24 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 25 Normal: mean = 25, VAR = 25 0.2 f 0.15 0.1 0.05 x 10 20 30 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 25 Normal: mean = 25, VAR = 25 0.1 f Change scale 0.08 0.06 0.04 0.02 x 10 20 30 40 50 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 30 Normal: mean = 30, VAR = 30 0.1 f 0.08 0.06 0.04 0.02 x 10 20 30 40 50 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 35 Normal: mean = 35, VAR = 35 0.1 f 0.08 0.06 0.04 0.02 x 10 20 30 40 50 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 40 Normal: mean = 40, VAR = 40 0.1 f 0.08 0.06 0.04 0.02 x 10 20 30 40 50 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 40 Normal: mean = 40, VAR = 40 0.1 f Change scale again 0.08 0.06 0.04 0.02 x 10 20 30 40 50 60 70 80 90 100 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 50 Normal: mean = 50, VAR = 50 0.1 f 0.08 0.06 0.04 0.02 x 10 20 30 40 50 60 70 80 90 100 Approximating distributions: Nannestad

Approximating distributions: Nannestad Poisson:  = 75 Normal: mean = 75, VAR = 75 0.1 f 0.08 0.06 0.04 0.02 x 10 20 30 40 50 60 70 80 90 100 Approximating distributions: Nannestad

Approximating distributions: Nannestad Summary: when can we? Binomial distribution Normal distribution Poisson distribution np  5 and n(1-p)  5 p is too small for the tables n large  > 20 Approximating distributions: Nannestad

Approximating distributions: Nannestad Summary: how do we? Binomial distribution Normal distribution  = np Poisson distribution Approximating distributions: Nannestad

Approximating distributions: Nannestad The end. Approximating distributions: Nannestad