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ELEMENTARY
STATISTICS
Chapter 4
Probability Distributions
EIGHTH
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
EDITION
MARIO F. TRIOLA
1
Chapter 4
Probability Distributions
4-1 Overview
4-2 Random Variables
4-3 Binomial Probability Distributions
4-4 Mean, Variance, Standard Deviation
for the Binomial Distribution
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
4-1
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 describe
what will probably happen instead of
what actually did happen.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Combining Descriptive Statistics Methods and
Probabilities to Form a Theoretical Model of
Behavior
Figure 4-1
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
4-2
Random Variables
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
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 graph, table, or formula that gives the
probability for each value of the random variable
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Table 4-1
Probability Distribution
Number of Girls Among Fourteen Newborn Babies
x
P(x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.000
0.001
0.006
0.022
0.061
0.122
0.183
0.209
0.183
0.122
0.061
0.022
0.006
0.001
0.000
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Probability Histogram
Figure 4-3
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Definitions
Discrete random variable
has 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.
Continuous random variable
has infinitely many values, and those values can
be associated with measurements on a
continuous scale with no gaps or interruptions.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Requirements for
Probability Distribution
The sum of all the probabilities of the distribution equals 1.
ΣP(x) = 1
where x assumes all possible values
All probabilities of the distribution must fall between 0 and 1
inclusive.
0  P(x)  1
for every value of x
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Mean, Variance and Standard Deviation
of a Probability Distribution
Mean
µ =  [x • P(x)]
Variance
2
2
 =  [(x - µ) • P(x)]
Standard Deviation
=
2
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Roundoff Rule for µ,  , and 
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, 2, and  to one
decimal place.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Example
A 4 point pop quiz was given and the scores
ranged from 0 to 4, with the corresponding
probabilities: 0.05, 0.2, 0.25, 0.3, 0.2
• Write the probability distribution in a table.
• Verify whether a probability distribution is
given.
• Compute the mean, variance and standard
deviation of the probability distribution.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Solution: Write the probability distribution in a table.
X (Quiz Score)
P(x)
0
0.05
1
0.2
2
0.25
3
0.3
4
0.2
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Solution: Verify whether a
probability distribution is given
• Yes it is a probability distribution because the sum of P(x) for all
values of x is 1 and P(x) for all values of x is between 0 and 1,
inclusive.
X (Quiz Scores)
P(x)
0
0.05
1
0.2
2
0.25
3
0.3
4
0.2
Sum
1.0
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Solution: Compute the mean
• Enter x values into L1, Enter P(x) into L2.
• Compute the products L3 = L1* L2.
• Find the sum(L3) which is the Mean: μ = 2.4
L1
L2
L3=L1*L2
x
P(x)
x * P(x)
0
0.05
0
1
0.2
0.2
2
0.25
0.5
3
0.3
0.9
4
0.2
0.8
Sum
1.0
2.4
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Solution: Compute the variance and standard deviation.
•
•
•
•
Having already found the mean of 2.4
Compute L4 = (L1- 2.4)2* L2.
Find the sum(L4) which is the 2 = 1.34  1.3
Find the  which is the square root of 2  =
= 1.16  1.2
1.34
L1
L2
L3 = L1*L2
L4=(L1-2.4)2*L2
x
P(x)
x * P(x)
(x- μ)2*P(x)
0
0.05
0
(0-2.4)2*0.05 = 0.288
1
0.2
0.2
(1-2.4)2*0.2 = 0.392
2
0.25
0.5
(2-2.4)2*0.25 = 0.040
3
0.3
0.9
(3-2.4)2*0.3 = 0.108
4
0.2
0.8
(4-2.4)2*0.2 = 0.512
Sum
1.0
2.4
1.34
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Definition
Expected Value
The average(mean) value of outcomes, if
the trials could continue indefinetly.
E =  [x • P(x)]
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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The Daily Number allows you to
place a bet that the three-digit
number of your choice. It cost $1 to
place a bet in order to win $500.
What is the expected value of gain or
loss?
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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E =  [x • P(x)]
Event
x
Win
$499
Lose
- $1
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
E =  [x • P(x)]
Event
x
P(x)
Win
$499
0.001
Lose
- $1
0.999
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
E =  [x • P(x)]
Event
x
P(x)
x • P(x)
Win
$499
0.001
0.499
Lose
- $1
0.999
- 0.999
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
E =  [x • P(x)]
Event
x
P(x)
x • P(x)
Win
$499
0.001
0.499
Lose
- $1
0.999
- 0.999
E = -$.50
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Expected Value
• This means that in the long run, for
each $1 bet, we can expect to lose an
average of $.50.
• In actuality a player either loses $1 or
wins $499, there will never be a loss of
$.50.
• If an expected value is $0, that means
that the game is fair, and favors no
side of the bet.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Alternate Solution:
Excluding the cost of playing.
Event
x
P(x)
x • P(x)
Win
$500
0.001
0.50
Lose
0
0.999
0
E = $.50, Which means the average win is $.50.
If the game cost $.50 to play E = 0 and it would be a “fair” game and
not favor either side of the bet.
If the game cost $.75 to play E = -.25 and it would not be a “fair”
game and would favor the house.
Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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