Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Probability Distributions…

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



OverviewThis chapter will deal with the

construction of

probability distributions by presenting possible outcomes along

with relative frequencies we expect.



Chapter 4Probability Distributions

4-1* Overview

4-2 Random Variables

4-3 & 4-4 Binomial Experiments

4-5* The Poisson Distribution



4-2Random Variables



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



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.



Definitions Probability Distribution

gives the probability for each value of the random variable



Probability Distribution for Number of USAir Crashes Among Seven

01234567

0.2100.3670.2750.1150.0290.0040+0+

x P(x)

Table 4-1



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



Requirements for Probability Distribution




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




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

0 P(x) 1 for every value of x



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



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



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.



DefinitionExpected Value

The average value of outcomes

E = [x • P(x)]

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Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Probability Distributions…