1. Chapter 5
Created by Bethany Stubbe and Stephan Kogitz

2. Discrete Probability Distributions
Chapter Five
Discrete Probability Distributions
GOALS
When you have completed this chapter, you will be able to:
ONE Define the terms random variable and probability distribution.
TWO Distinguish between a discrete and continuous probability distributions.
THREE Calculate the mean, variance, and standard deviation of a discrete probability distribution.

3. Discrete Probability Distributions
Chapter Five continued
Discrete Probability Distributions
GOALS
When you have completed this chapter, you will be able to:
FOUR Describe the characteristics and compute probabilities using the binomial probability distribution.
FIVE Describe the characteristics and compute probabilities using the hypergeometric distribution.
SIX Describe the characteristics and compute the probabilities using the Poisson distribution.

4. Random Variables
A random variable whose value is determined by the outcome of a random experiment.
A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability.

5. Types of Probability Distributions
A discrete probability distribution can assume only certain outcomes.
A continuous probability distribution can assume an infinite number of values within a given range.

6. Types of Probability Distributions
Examples of a discrete distribution are:
The number of students in a class.
The number of children in a family.
Number of home mortgages approved by a bank last week.

7. Types of Probability Distributions
Examples of a continuous distribution include:
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.

8. Features of a Discrete Distribution
The main features of a discrete probability distribution are:
The sum of the probabilities of the various outcomes is 1.00.
The probability of a particular outcome is between 0 and 1.00.
The outcomes are mutually exclusive.

9. Example 1
Consider a random experiment in which a coin is tossed three times. Let x be the number of heads. Let H represent the outcome of a head and T the outcome of a tail.

10. EXAMPLE 1 continued
The possible outcomes for such an experiment will be:
TTT, TTH, THT, THH,
HTT, HTH, HHT, HHH.
Thus the possible values of x (number of heads) are 0,1,2,3.

11. EXAMPLE 1 continued The outcome of zero heads occurred once.
The outcome of one head occurred three times.
The outcome of two heads occurred three times.
The outcome of three heads occurred once.
From the definition of a random variable, x as defined in this experiment, is a random variable.

12. The Mean of a Discrete Probability Distribution
reports the central location of the data.
is the long-run average value of the random variable.
is also referred to as its expected value, E(X), in a probability distribution.
is a weighted average.

13. The Mean of a Discrete Probability Distribution
The mean is computed by the formula:
where μ is the mean and P(x) is the probability of the various outcomes x.

14. The Variance of a Discrete Probability Distribution
The variance measures the amount of spread (variation) of a distribution.
The variance of a discrete distribution is denoted by the Greek letter σ2 (sigma squared).
The standard deviation is the square root of σ2.

15. The Variance of a Discrete Probability Distribution
The variance of a discrete probability distribution is computed from the formula:

16. EXAMPLE 2
Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week:

17. EXAMPLE 2 continued

18. EXAMPLE 2 continued
Compute the mean number of houses painted per week:

19. EXAMPLE 2 continued
Compute the variance of the number of houses painted per week:

20. Binomial Probability Distribution
The binomial distribution has the following characteristics:
An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure.
The data collected are the results of counts.
The probability of success stays the same for each trial.
The trials are independent.

21. Binomial Probability Distribution
To construct a binomial distribution, let
n be the number of trials
x be the number of observed successes
π be the probability of success on each trial

22. Binomial Probability Distribution
The formula for the binomial probability distribution is:

23. EXAMPLE 3 Exactly three were unemployed.
Statistics Canada reports that as of 2001, approximately 20% of the workforce in Gaspésie-Îles-de-la-Madeleine in Quebec was unemployed. From a sample of 14 workers, calculate the following probabilities:
Exactly three were unemployed.
At least three were unemployed.
At least none were unemployed.

24. EXAMPLE 3 continued The probability of exactly 3:
The probability of at least 3 is:

25. Example 3 continued
The probability of at least one being unemployed.

26. Mean & Variance of the Binomial Distribution
The mean is found by:
The variance is found by:

27. EXAMPLE 4 From EXAMPLE 3, recall that π =.2 and n=14.
Hence, the mean is:
μ= n π = 14(.2) = 2.8.
The variance is:
σ2 = n (1- π ) = (14)(.2)(.8)
=2.24.

28. Finite Population The number of students in this class.
A finite population is a population consisting of a fixed number of individuals, objects, or measurements. Examples include:
The number of students in this class.
The number of cars in the parking lot.
The number of homes built in Toronto.

29. Hypergeometric Distribution
The hypergeometric distribution has the following characteristics:
There are only 2 possible outcomes.
The probability of a success is not the same on each trial.
It results from a count of the number of successes in a fixed number of trials.

30. Hypergeometric Distribution
The formula for finding a probability using the hypergeometric distribution is:
where N is the size of the population, S is the number of successes in the population, x is the number of successes in a sample of n observations.

31. Hypergeometric Distribution
Use the hypergeometric distribution to find the probability of a specified number of successes or failures if:
the sample is selected from a finite population without replacement
the size of the sample n is greater than 5% of the size of the population N

32. EXAMPLE 5
Transport Canada has a list of 10 reported safety violations. Suppose only 4 of the reported violations are actual violations and Transport Canada will only be able to investigate five of the violations. What is the probability that three of five violations randomly selected to be investigated are actually violations?

33. EXAMPLE 5 continued

34. Poisson Probability Distribution
The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.
The limiting form of the binomial distribution where the probability of success π is small and n is large is called the Poisson probability distribution.

35. Poisson Probability Distribution
The Poisson distribution can be described mathematically using the formula:
where μ is the mean number of successes in a particular interval of time, e is the constant , and x is the number of successes.

36. Poisson Probability Distribution
The mean number of successes μ can be determined in binomial situations by n π, where n is the number of trials and π the probability of a success.
The variance of the Poisson distribution is also equal to n π.

37. EXAMPLE 6
The University Health Clinic facility specializes in caring for minor injuries, colds, and flu. For the evening hours of 6-10 PM the mean number of arrivals is 4.0 per hour. What is the probability of 4 arrivals in an hour?