Probability Distributions. Chapter 6
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1 Probability Distributions Chapter 6 McGraw-Hill/Irwin The McGraw-Hill Companies, Inc. 2008
2 Types of Random Variables Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement 2
3 Discrete Random Variables - Examples The number of students in a class. The number of children in a family. The number of cars entering a carwash in a hour. Number of home mortgages approved by Coastal Federal Bank last week. 3
4 Continuous Random Variables - Examples The distance students travel to class. The time it takes an executive to drive to work. The length of an afternoon nap. The length of time of a particular phone call. 4
5 Features of a Discrete Distribution The main features of a discrete probability distribution are: The sum of the probabilities of the various outcomes is The probability of a particular outcome is between 0 and The outcomes are mutually exclusive. 5
6 The Mean of a Probability Distribution MEAN The mean is a typical value used to represent the central location of a probability distribution. The mean of a probability distribution is also referred to as its expected value. 6
7 The Variance, and Standard Deviation of a Probability Distribution Variance and Standard Deviation Measures the amount of spread in a distribution The computational steps are: 1. Subtract the mean from each value, and square this difference. 2. Multiply each squared difference by its probability. 3. Sum the resulting products to arrive at the variance. The standard deviation is found by taking the positive square root of the variance. 7
8 Mean, Variance, and Standard Deviation of a Probability Distribution - Example John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday. 8
9 9 Mean of a Probability Distribution - Example
10 10 Variance and Standard Deviation of a Probability Distribution - Example
11 Binomial Probability Distribution Characteristics of a Binomial Probability Distribution There are only two possible outcomes on a particular trial of an experiment. The outcomes are mutually exclusive, The random variable is the result of counts. Each trial is independent of any other trial 11
12 Binomial probability distribution The probability can be computed as: = 1, =0,1,, : number of trial, p: prob. : n choose x : use calculator Mean== Variance= =(1 ) Note that: sometimes used: = 12
13 Binomial Probability - Example There are five flights daily from Pittsburgh via US Airways into the Bradford, Pennsylvania, Regional Airport. Suppose the probability that any flight arrives late is.20. What is the probability that none of the flights are late today? 13
14 Binomial Dist. Mean and Variance: Example For the example regarding the number of late flights, recall that π =.20 and n = 5. What is the average number of late flights? What is the variance of the number of late flights? 14
15 Poisson Probability Distribution The Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume. Assumptions of the Poisson Distribution (1) The probability is proportional to the length of the interval. (2) The intervals are independent. 15
16 Poisson Probability distribution Mean=variance= The probability: =!, where =0,1,2, Not that mean== 16
17 Poisson Probability Distribution - Example Assume baggage is rarely lost by Northwest Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags. 17
18 18 End of Chapter 6
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