Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Probability Distributions…
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 1
Probability Probability DistributionsDistributions
Chapter 4Chapter 4
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Addison Wesley Longman 2
OverviewThis chapter will deal with the
construction of
probability distributions by presenting possible outcomes along
with relative frequencies we expect.
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Chapter 4Probability Distributions
4-1* Overview
4-2 Random Variables
4-3 & 4-4 Binomial Experiments
4-5* The Poisson Distribution
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4-2Random Variables
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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
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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.
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Definitions Probability Distribution
gives the probability for each value of the random variable
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Probability Distribution for Number of USAir Crashes Among Seven
01234567
0.2100.3670.2750.1150.0290.0040+0+
x P(x)
Table 4-1
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Probability Histogram0.40
0.30
0.20
0.10
00 1 2 3 4 5 6 7
Probability Histogram Number of USAir Crashes Among SevenFigure 4-3
Number of USAir Crashes Among Seven
Prob
abili
ty
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Requirements for Probability Distribution
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Requirements for Probability Distribution
P(x) = 1 where x assumes all possible values
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Requirements for Probability Distribution
P(x) = 1 where x assumes all possible values
0 P(x) 1 for every value of x
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Formula 4-1 Mean: µ = x • P(x)
Formula 4-2Variance: 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
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Mean, Variance and Standard Deviation of a Probability
DistributionFormula 4-1
Mean: µ = x • P(x)
Formula 4-2Variance: 2 = [(x – µ)2
• P(x)]
2 = [ x2
• P(x)] – µ 2 (shortcut)
Formula 4-4 SD: = [ x 2 • P(x)]–µ 2
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Round off 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.
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DefinitionExpected Value
The average value of outcomes
E = [x • P(x)]
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