1Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSChapter 4 Probability Distributions

2Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Chapter 4

Probability Distributions4-1 Overview

4-2 Random Variables

4-3 Binomial Probability Distributions

4-4 Mean, Variance, Standard Deviation for the Binomial Distribution

4-5 The Poisson Distribution

3Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

4-1 OverviewThis chapter will deal with the construction of

probability distributions

by combining the methods of Chapter 2 with the those of Chapter 3.

Probability Distributions will describe what will probably happen instead of what

actually did happen.

4Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 4-1

Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of

Behavior

5Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

4-2

Random Variables

6Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definitions Random Variable

a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure

Probability Distribution

a graph, table, or formula that gives the probability for each value of the random variable

7Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Probability DistributionNumber of Girls Among Fourteen Newborn Babies

0 1 2 3 4 5 6 7 8 91011121314

0.0000.0010.0060.0220.0610.1220.1830.2090.1830.1220.0610.0220.0060.0010.000

x P(x)

Table 4-1

8Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionsDiscrete random variable

has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process.

Continuous random variable

has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions.

9Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 4-3

Probability Histogram

10Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Requirements for Probability Distribution

P(x) = 1 where x assumes all possible values

11Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Requirements for Probability Distribution

P(x) = 1 where x assumes all possible values

0 P(x) 1 for every value of x

12Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Formula 4-1 µ = [x • P(x)]

Formula 4-2

2 = [(x - µ)2 • P(x)]

Formula 4-3

2 = [ x2

• P(x)] - µ 2 (shortcut)

Mean, Variance and Standard Deviation of a Probability Distribution

13Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Formula 4-1 µ = [x • P(x)]

Formula 4-2

2 = [(x - µ)2 • P(x)]

Formula 4-3

2 = [ x2

• P(x)] - µ 2 (shortcut)

Formula 4-4 = [ x 2 • P(x)] - µ 2

Mean, Variance and Standard Deviation of a Probability Distribution

14Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Formula 4-1 µ = [x • P(x)]

Formula 4-2

2 = [(x - µ)2 • P(x)]

Formula 4-3

2 = [ x2

• P(x)] - µ 2 (shortcut)

Formula 4-4 = [ x 2 • P(x)] - µ 2

Mean, Variance and Standard Deviation of a Probability Distribution

15Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Roundoff Rule for µ, 2, and

Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, 2, and to one decimal place.

16Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definition

Expected Value The average value of outcomes

E = [x • P(x)]

17Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

E = [x • P(x)]

Event

Win

Lose

18Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

E = [x • P(x)]

Event

Win

Lose

x

$499

- $1

19Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

E = [x • P(x)]

Event

Win

Lose

x

$499

- $1

P(x)

0.001

0.999

20Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

E = [x • P(x)]

Event

Win

Lose

x

$499

- $1

P(x)

0.001

0.999

x • P(x)

0.499

- 0.999

21Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

E = [x • P(x)]

Event

Win

Lose

x

$499

- $1

P(x)

0.001

0.999

x • P(x)

0.499

- 0.999

E = -$.50