Probability Distributions
Chapter 4
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison
Wesley
Longman
Addison
Wesley Longman
1
Overview
This chapter will deal with the
construction of
probability distributions
by presenting possible outcomes along
with relative frequencies we expect.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Chapter 4
Probability Distributions
4-1* Overview
4-2 Random Variables
4-3 & 4-4 Binomial Experiments
4-5* The Poisson Distribution
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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4-2
Random Variables
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Example: 10 balls marked 0 to 9 and
placed in a box. Pick one ball out from the
box.
Q: How to represent the outcome (i.e., the
number on that ball)?
Solution:
Use a variable, say x, to represent
the outcome ----- x is called a random variable
Two meanings:
(1) x is one of the 10 possible outcomes: 0,1, …, 9
(2) Each can happen with a positive chance
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Definitions
 Random Variable
a variable (usually x) that has a single numerical
value (determined by chance) for each outcome
of an experiment
Discrete random variables have a finite
number or countable number of values.
Continuous random variables have infinitely
many values which can be associated with
measurements on a continuous scale with no
gaps or interruptions.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Definitions
 Probability Distribution
gives the probability for each value of the
random variable
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Table 4-1
Probability Distribution for Number
of USAir Crashes Among Seven
x
P(x)
0
1
2
3
4
5
6
7
0.210
0.367
0.275
0.115
0.029
0.004
0+
0+
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Probability Histogram
0.40
Probability
0.30
0.20
0.10
0
0
1
2
3
4
5
6
7
Number of USAir Crashes Among Seven
Figure 4-3
Probability Histogram Number of USAir Crashes Among Seven
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Requirements for
Probability Distribution
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Requirements for
Probability Distribution
S P(x) = 1
where x assumes all possible values
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Requirements for
Probability Distribution
S P(x) = 1
where x assumes all possible values
0  P(x)  1
for every value of x
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Mean, Variance and
Standard Deviation of a
Probability Distribution
Formula 4-1
µ = S x • P(x)
Mean:
Formula 4-2
Variance: s = S [(x – µ) • P(x)]
2
2
Formula 4-3
2
2
2
s = [S x • P(x)] – µ (shortcut)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Mean, Variance and
Standard Deviation of a
Probability Distribution
Formula 4-1
µ = S x • P(x)
Mean:
Formula 4-2
Variance: s = S [(x – µ) • P(x)]
2
2
2
s = [S x • P(x)] – µ (shortcut)
2
2
Formula 4-4
SD:
s=
[S x 2 • P(x)]–µ 2
Copyright © 1998, Triola, Elementary Statistics
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Round off Rule for µ, s , and s
2
• 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, s 2, and s to one
decimal place.
Copyright © 1998, Triola, Elementary Statistics
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Definition
Expected Value
The average value of outcomes
E = S [x • P(x)]
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