Transcript Ch 5.1, 5.2 - San Diego Mesa College

Overview
Created by Tom Wegleitner, Centreville, Virginia
Edited by Olga Pilipets, San Diego, California
Copyright © 2007 Pearson
Education, Inc Publishing as
Pearson Addison-Wesley.
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This chapter will deal with the construction of
discrete probability distributions
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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Random Variables
Created by Tom Wegleitner, Centreville, Virginia
Edited by Olga Pilipets, San Diego, California
Copyright © 2007 Pearson
Education, Inc Publishing as
Pearson Addison-Wesley.
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
Random variable
a variable (typically represented by x) that has
a single numerical value, determined by
chance, for each outcome of a procedure
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
Discrete random variable
Random variable x takes on counting (natural) values, where
“counting” refers to the fact that there might be infinitely many
values, but they result from a counting process
 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
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We could flip a coin 3 times. There are 8
possible outcomes;
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
If we are interested in the number of
heads that appear during the three flips
we could get any of the following
numbers: 0, 1, 2, or 3.
The numbers 0, 1, 2, and 3, are the values
of a random variable that has been
associated with the possible outcomes.
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 Probability
distribution
a listing of all possible values that the variable
can assume along with their corresponding
probabilities.
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A coin is flipped 3 times giving the following
sample space
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
We are interested in the number of heads that
appear during the three flips. Assigning a
probability value to each possible random
variable we construct the following probability
distribution.
x
0
1
2
3
P(x)
1/8
3/8
3/8
1/8
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a) The cost of conducting a genetics experiment.
b) The number of supermodels who ate pizza
yesterday.
c) The exact life span of a kitten.
d) The number of statistic professors who read
a newspaper this morning
e) The weight of a feather.
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0  P(x)  1
for every individual value of x.
 P(x) = 1
where x assumes all possible values.
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A researcher reports that when groups of
four children are randomly selected from a
population of couples meeting certain
criteria, the probability distribution for the
number of boys is given in the
accompanying table.
x
0
1
2
3
4
P(x)
0.502
0.365
0.098
0.011
0.001
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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)
2
2
2
 [x 2 • P(x)] – µ 2
Deviation
=
Standard
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x
0
1
2
3
4
P(x)
0.502
0.365
0.109
0.023
0.001
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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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Example
Twelve jurors are to be randomly selected from a
population in which 80% of the jurors are MexicanAmerican. If we assume that jurors are randomly
selected without bias, and if we let
x = the number of Mexican-American jurors among 12
jurors,
we will get a probability distribution represented by the
following table:
Cont-d
x (MexicanAmericans)
0
1
2
3
P(x)
0+
0+
0+
0+
x (MexicanAmericans)
P(x)
7
8
9
4
5
6
0.001 0.003 0.016
10
11
12
0.053 0.133 0.236Copyright
0.283
0.206
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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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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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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
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
In a study of brand recognition of Sony,
groups of four consumers are interviewed. If
x is the number of people in the group who
recognize the Sony brand name, then x can
be:
0, 1, 2, 3, or 4
and the corresponding probabilities are 0.0016, 0.0250,
0.1432, 0.3892, and 0.4096
Is it unusual to randomly select four consumers and find
that none of them recognize the brand name of Sony?
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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)]
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When you give a casino $5 for a bet on the
“pass line” in a casino game of dice, there is a
251/495 probability that you will lose $5 and
there is a 244/495 probability that you will
make a net gain of $5. (If you win, the casino
gives you $5 and you get to keep your $5
bet, so the net gain is $5.) What is your
expected value? In the long run, how much
do you lose for each dollar bet?
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The CAN Insurance Company charges a 21-yearold male a premium of $250 for a one-year
$100,000 life insurance policy. A 21-yearold male has a 0.9985 probability of living for
a year.
a) From the perspective of a 21-year-old male
(or his estate), what are the values of the two
different outcomes?
b) What is the expected value for a 21-year-old
male who buys the insurance?
c) What would be the cost of the insurance policy
if the company just breaks even (in the long
run with many such policies), instead of
making a profit?
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Recap
In this section we have discussed:
 Random variables and probability distributions.
 Probability histograms.
 Requirements for a probability distribution.
 Mean, variance and standard deviation of a
probability distribution.
 Expected value.
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