1. probability distributions
Chapter 5

2. Outline Section 5-1: Expected Value Section 5-2: Binomial Distribution
Section 5-3: Poisson Distribution (OMIT)
Section 5-4: Hypergeometric Distribution (OMIT)

3. Introduction

4. Overview
This chapter will deal with the construction of probability distributions by combining methods of descriptive statistics from Chapters 2 and 3 and those of probability presented in Chapter 4.
A probability distribution, in general, will describe what will probably happen instead of what actually did happen

5. Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.
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6. Why do we need probability distributions?
Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results
Saleswoman can compute probability that she will make 0, 1, 2, or 3 or more sales in a single day. Then, she would be able to compute the average number of sales she makes per week, and if she is working on commission, she will be able to approximate her weekly income over a period of time.
An investor wants to compare the risks of two different stock options for his portfolio

7. Probability Distributions

8. Remember
From Chapter 1, a variable is a characteristic or attribute that can assume different values
Represented by various letters of the alphabet
From Chapter 1, a random variable is a variable whose values are determined by chance
Typically assume values of 0,1,2…n

9. Remember Can be assigned values such as 0, 1, 2, 3 “Countable”
Discrete Variables (Data)—
Chapter 5
Continuous Variables (Data)--- Chapter 6
Can be assigned values such as 0, 1, 2, 3
“Countable”
Examples:
Number of children
Number of credit cards
Number of calls received by switchboard
Number of students
Can assume an infinite number of values between any two specific values
Obtained by measuring
Often include fractions and decimals
Examples:
Temperature
Height
Weight
Time

10. Examples: State whether the variable is discrete or continuous
The height of a randomly selected giraffe living in Kenya
The number of bald eagles located in New York State
The exact time it takes to evaluate
The number of textbook authors now sitting at a computer
The exact life span of a kitten
The number of statistics students now reading a book
The weight of a feather

11. Discrete Probability Distribution
Consists of the values a random variable can assume and the corresponding probabilities of the values.
The probabilities are determined theoretically or by observation
Can be shown by using a graph (probability histogram), table, or formula
Two requirements:
The probability of each event in the sample space must be between or equal to 0 and 1. That is, 0 < P(x) < 1
The sum of the probabilities of all the events in the sample space must equal 1; that is, SP(x) = 1

12. Example: Determine whether the distribution represents a probability distribution. If it does not, state why. 8) 9) x 3 6 8 12
P(x)
0.3
0.5
0.7
-0.8 x 1 2 3 4 5
P(x)
0.3
0.1
0.2

13. Example: Determine whether the distribution represents a probability distribution. If it does not, state why.
A researcher reports that when groups of four children are randomly selected from a population of couples meeting certain criteria, the probability distribution for the number of girls is given in the accompanying table x
P(x)
0.502 1
0.365 2
0.098 3
0.011 4
0.001

14. Section 5.1 Expected Value
Objective: Calculate the expected value of a probability distribution

15. Once we know that a probability distribution exist, we can describe it using various descriptive statistics
Visually using a graph, table, or formula
Algebraically, we can find the mean, variance, and standard deviation

16. Mean of a general discrete probability distribution
m= population mean since ALL possible values are considered Mean is also known as “Expected Value” Mean should be rounded to one more decimal place than the outcome x. Always simplify fractions

17. Variance & standard deviation

18. Example –Use table on example 9 to find mean and standard deviation1 2 3 4 5
P(x)
0.3
0.1
0.2

19. Assignment
Worksheet ---Section 5.1

20. Answers to Examples Continuous Discrete
This is not a probability distribution because one of the probabilities is negative (is not between 0 and 1)