Chapter 5
Probability Distributions
5-1 Overview
5-2 Random Variables
5-3 Binomial Probability Distributions
5-4 Mean, Variance and Standard Deviation
for the Binomial Distribution
5-5 The Poisson Distribution
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Section 5-1
Overview
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Overview
This chapter will deal with the construction of
discrete probability distributions
by combining the methods of descriptive
statistics presented in Chapter 2 and 3 and
those of probability presented in Chapter 4.
Probability Distributions will describe what
will probably happen instead of what
actually did happen.
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Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions
by presenting possible outcomes along with the relative
frequencies we expect.
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Section 5-2
Random Variables
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Key Concept
This section introduces the important concept of a
probability distribution, which gives the probability for
each value of a variable that is determined by chance.
Give consideration to distinguishing between
outcomes that are likely to occur by chance and
outcomes that are “unusual” in the sense they are not
likely to occur by chance.
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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 description that gives the probability for
each value of the random variable; often
expressed in the format of a graph, table, or
formula
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Example
12 Jurors are to be randomly selected from a population in
which 80% of the jurors are Mexican-American. If we
assume that jurors are randomly selected without bias,
here is an example of a probability distribution depicting
these probabilities (let x = number of Mexican-American
jurors among 12 jurors). Probabilities that are very small
(such as 0.000000123) are represented by 0+.
x
0
1
2
P(x)
0+
0+ 0+
3
4
5
6
7
8
9
10
11
12
0+
0.001
0.003
0.016
0.053
0.133
0.236
0.283
0.206
0.069
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Graphs
The probability histogram is very similar to a relative
frequency histogram, but the vertical scale shows
probabilities.
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Definitions
 Discrete random variable
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. Example: Count of number of
movie patrons
 Continuous random variable
infinitely many values, and those values can
be associated with measurements on a
continuous scale in such a way that there
are no gaps or interruptions. Example: The
measured voltage of a smoke detector
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battery.
Continuous or discrete?
1) The number of eggs a hen lays
2) The count of the number of stats students
present in class today
3) The amount of milk a cow produces in one
day
4) The measure of voltage for a particular
smoke detector battery
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Requirements for
Probability Distribution
 P(x) = 1
where x assumes all possible values.
0  P(x)  1
for every individual value of x.
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Example
• Does this table describe a probability
distribution?
X
0
1
2
3
P(x)
0.2
0.5
0.4
0.3
• Does P(x) = x/3 (where x can be 0, 1, or 2)
determine a probability distribution?
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Mean, Variance and
Standard Deviation of a
Probability Distribution
µ =  [x • P(x)]
Mean
 =  [(x – µ) • P(x)]
Variance
 = [ x2 • P(x)] – µ 2
Variance (shortcut)
 =  [x 2 • P(x)] – µ 2
Standard Deviation
2
2
2
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Roundoff Rule for
2
µ, , 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, , and 2 to one decimal
place.
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Identifying Unusual Results
Range Rule of Thumb
According to the range rule of thumb, most
values should lie within 2 standard deviations
of the mean.
We can therefore identify “unusual” values by
determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
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The table below describes the prob. dist. for the number of MexicanAmericans among 12 randomly selected jurors in Hidalgo County, Texas.
Assuming that we repeat the process of randomly selecting 12 jurors and
counting the number of Mexican-Americans each time, find the mean
number of Mexican-Americans (among 12), the variance, and the std. dev.
X
P(x)
x*P(x)
X-sq
X-sq* P(x)
0
0+
0.000
0
0.000
1
0+
0.000
1
0.000
2
0+
0.000
4
0.000
3
0+
0.000
9
0.000
4
0.001
0.004
16
0.000
5
0.003
0.015
25
0.075
6
0.016
0.096
36
0.576
7
0.053
0.371
49
2.597
8
0.133
1.064
64
8.512
9
0.236
2.124
81
19.116
0.
0.283
2.830
100
28.300
11
0.206
2.266
121
24.926
12
0.069
0.828
144
9.936
Total
9.598
94.054
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Identifying Unusual Results
Probabilities
Rare Event Rule
If, under a given assumption (such as the
assumption that a coin is fair), the probability of a
particular observed event (such as 992 heads
in 1000 tosses of a coin) is extremely small, we
conclude that the assumption is probably not
correct.
 Unusually high: x successes among n trials is an
unusually high number of successes if P(x or
more) ≤ 0.05.
 Unusually low: x successes among n trials is an
unusually low number of successes if P(x or
fewer) ≤ 0.05.
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Example
If 80% of those eligible for jury duty in Hidalgo
County are Mexican-American, then a jury of
12 randomly selected people should have
around 9 or 10 who are Mexican-American. Is
7 Mexican-American jurors among 12 an
unusually low number? Does the selection of
only 7 Mexican-Americans among 12 jurors
suggest that there is discrimination in the
selection process?
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Definition
The expected value of a discrete random
variable is denoted by E, and it represents
the average value of the outcomes. It is
obtained by finding the value of  [x • P(x)].
E =  [x • P(x)]
The mean of a discrete random variable is
The same as its expected value!
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Example
If you bet $1 in Kentucky’s Pick 4 Lottery game,
you either lose $1 or gain $4999 (the winning
prize is $5000, but your $1 bet is not returned,
so the net gain is $4999). The game is played by
selected a four-digit number between 0000 and
9999. If you bet $1 on 1234, what is your
expected value of gain or loss?
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You do!
The random variable x is a count of the number
of girls that occur when two babies are born.
Construct a table representing the probability
distribution, then find its mean and standard
deviation.
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Recap
In this section we have discussed:
 Combining methods of descriptive statistics with
probability.
 Random variables and probability distributions.
 Probability histograms.
 Requirements for a probability distribution.
 Mean, variance and standard deviation of a
probability distribution.
 Identifying unusual results.
 Expected value.
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