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Chapter 4
Probability Distributions4-1 Overview4-2 Random Variables4-3 Binomial Probability Distributions4-4 Mean, Variance, Standard Deviationfor the Binomial Distribution
Overview
This 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 describewhat will
probably
happen instead of
what actually
did^ happen.
Combining Descriptive Statistics Methods andProbabilities to Form a Theoretical Model of Figure 4-
Behavior
Random Variables
Definitions
^ Random Variablea variable (typically represented by
x ) that has a
single numerical value, determined by chance,for each outcome of a procedure Probability Distributiona graph, table, or formula that gives theprobability for each value of the random variable
Probability Distribution Number of Girls Among Fourteen Newborn Babies
(^01234567891011121314)
0.0000.0010.0060.0220.0610.1220.1830.2090.1830.1220.0610.0220.0060.0010. x^
P ( x )
Table 4-
Definitions
Discrete random variablehas either a finite number of values or countablenumber of values, where ‘countable’ refers to thefact that there might be infinitely many values,but they result from a counting process. Continuous random variablehas infinitely many values, and those values canbe associated with measurements on acontinuous scale with no gaps or interruptions.
Figure 4-
Probability Histogram
minimum
=^ μ^ - 2(
σ)
maximum
=^ μ^ + 2(
σ)
Usual Sample Values
Probability Distribution Number of Girls Among Fourteen Newborn Babies
(^01234567891011121314)
0.0000.0010.0060.0220.0610.1220.1830.2090.1830.1220.0610.0220.0060.0010. x^
P ( x )
Table 4-
minimum
=^ 7.
maximum
=^ 7.
- 2(1.876) = 10. Usual Sample Values
Using the Rare Event RuleIf, under a given assumption (such as theassumption that boys and girls are equally likely),the probability of a particular observed event(such as 13 girls in 14 births) is extremely small,we conclude that the assumption was probablynot correct (boys and girls NOT equally likely; thegender selection technique did have an effect).
X^ is unusually high if with
x^ successes among
n
trials, P(x or more) is very small (such as 0.05 or
less)
X^ is unusually low if with
x^ successes among
n
trials, P(x or fewer) is very small (such as 0.05 or
less) Using Probabilities toDetermine When Results
Are Unusual
Probability Distribution Number of Girls Among Fourteen Newborn Babies
(^01234567891011121314)
0.0000.0010.0060.0220.0610.1220.1830.2090.1830.1220.0610.0220.0060.0010. x^
P ( x )
Table 4-
DefinitionExpected Value
The average value of outcomesE =
Σ^ [
x^ • P(
x )]
E =^ Σ
[x • P(x)] EventWinLose
x $499- $
P(x)0.0010.
