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Transcript of 1 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001....
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