©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin Probability Distributions Chapter 6

2 Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions. Calculate the mean, variance, and standard deviation of a discrete probability distribution. Describe the characteristics of and compute probabilities using the binomial probability distribution. Describe the characteristics of and compute probabilities using the hypergeometric probability distribution.- skip Describe the characteristics of and compute probabilities using the Poisson distribution. GOALS

3 Random Variables Random variable a quantity resulting from an experiment that, by chance, can assume different values. (eg) The number of head in three coin tosses.

4 What is a Probability Distribution? Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero, one, two, and three heads. What is the probability distribution for the number of heads?

5 Probability Distribution of Number of Heads Observed in 3 Tosses of a Coin

6 Characteristics of a Probability Distribution A random variable can be described by its probability distribution. Probability distribution is similar to the relative frequency distribution * difference: past vs future

7 Types of Random Variables Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something (not always). – Some may have decimal values (7.2, 8.9, --). Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement – Width, height, air pressure, etc. Probability distribution (discrete r.v.) & probability density function (continuous r.v.)

8 Discrete Random Variables - Examples The number of students in a class. The number of children in a family. The number of cars passing through a toll gate in an hour.

9 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.

10 Features of a Discrete Probability Distribution The main features of a discrete probability distribution are: The sum of the probabilities of the all outcomes is The probability of a particular outcome is between 0 and The outcomes are mutually exclusive.

11 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.

12 The Variance, and Standard Deviation of a Probability Distribution Variance and Standard Deviation Measures the degree of spread around its mean 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 the positive square root of the variance.

13 Mean, Variance, and Standard Deviation of a Probability Distribution (Random Variable)- Example John, a car salesman, 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.

14 Mean of a Probability Distribution (Random Variable) - Example

15 Variance and Standard Deviation of a Probability Distribution (Random Variable) - Example

16 Binomial Probability Distribution Characteristics of a Binomial Probability Distribution Outcome on each trial of an experiment is classified into one of two mutually exclusive categories. – ‘a success’ or ‘a failure’ The random variable counts the number of ‘successes’ in a fixed (total) number of trials. The probability of success is the same for each trial. Each trial is independent of any other trial. – The outcome of one trial does not affect the outcome of any other trial.

17 Binomial Probability Formula

18 Binomial Probability - Example There are five flights daily from Pittsburgh to Boston. Suppose the probability that any flight arrives late is.20. What is the probability that none (one) of the flights is late today?

19 Binomial Probability - Excel

20 Binomial Dist. – Mean and Variance

21 For the example regarding the number of late flights, recall that  =.20 and n = 5. What is the average number of late flights? What is the variance of the number of late flights? Binomial Dist. – Mean and Variance: Example

22 Binomial Dist. – Mean and Variance: Another Solution

23 Binomial Distribution- Table (pp ) Five percent of the gears produced by a milling machine are defective. What is the probability that out of six gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six?

24 Binomial – Shapes for Varying  (n constant) * how the shape changes?

25 Binomial – Shapes for Varying n (  constant) * how the shape changes?

26 Cumulative Binomial Probability Distributions A study by the Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?

27 Cumulative Binomial Probability Distributions - Excel

28 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, or area. Assumptions of the Poisson Distribution (1) The probability is proportional to the length of the interval. (2) The intervals do not overlap and are independent. Poisson distribution is a limiting form of the binomial distribution, when  is very small & n is very large.

29 Poisson Probability Distribution The Poisson distribution can be described mathematically using the formula:

30 Poisson Probability Distribution The mean number of successes  can be determined in binomial situations by n . – where n is the number of trials and  the probability of a success. – The variance of the Poisson distribution is also equal to n .

31 Assume baggage is rarely lost by Korea Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean of lost bags per flight is 0.3. If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags in a flight. Poisson Probability Distribution - Example

32 Poisson Probability Distribution - Table Suppose the number of lost bags per flight follows a Poisson distribution with mean = 0.3. Find the probability of losing no bag, one bag, two bags, ---, on a particular flight. (Table is available). * When should the supervisor be suspicious about lost bags?

33 Poisson distribution as an estimate of binomial distribution (large n, small  ) Example: probability of a strong typoon striking a certain area in any one year-.05 and homeowner purchased a house there on a 30 year mortgage. (Insurance company) Want to know the likelihood of at least one typoon strike during the 30 year period. Two approaches to calculate P(x ≥ 1) = 1 - P(x=0). – Poisson distribution: (using u = n  = 1.5) – Binomial distribution: (using  =.05, n = 30)

34 End of Chapter 6