Download JOHN S. LOUCKS Chapter 5 Discrete Probability Distributions

Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2005 Thomson/South-Western
Slide 1
Chapter 5
Discrete Probability Distributions
Random Variables
Discrete Probability Distributions
Expected Value and Variance
Binomial Distribution
Poisson Distribution
.40
Hypergeometric Distribution
.30
.20
.10
0
1
2
3
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Slide 2
Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
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Slide 3
1
Example: JSL Appliances
Discrete random variable with a finite number
of values
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
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Slide 4
Example: JSL Appliances
Discrete random variable with an infinite sequence
of values
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.
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Slide 5
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents
reported on tax return
Type
Discrete
Distance from x = Distance in miles from
home to store
home to the store site
Continuous
Own dog
or cat
Discrete
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
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Slide 6
2
Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.
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Slide 7
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
Σf(x) = 1
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Slide 8
Discrete Probability Distributions
Using past data on TV sales, …
a tabular representation of the probability
distribution for TV sales was developed.
Units Sold
0
1
2
3
4
Number
of Days
80
50
40
10
20
200
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x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
1.00
80/200
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3
Discrete Probability Distributions
Graphical Representation of Probability Distribution
Probability
.50
.40
.30
.20
.10
0
1
2
3
4
Values of Random Variable x (TV sales)
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Slide 10
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
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Slide 11
Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) = µ = Σxf(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) = σ 2 = Σ(x - µ)2f(x)
The standard deviation, σ, is defined as the positive
square root of the variance.
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Slide 12
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Expected Value and Variance
Expected Value
f(x)
xf(x)
.40
.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
x
0
1
2
3
4
expected number of
TVs sold in a day
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Slide 13
Expected Value and Variance
Variance and Standard Deviation
x
x-µ
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
(x - µ)2
f(x)
(x - µ)2f(x)
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
.576
.010
.128
.162
.784
Variance of daily sales = σ = 1.660
2
TVs
squared
Standard deviation of daily sales = 1.2884 TVs
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Slide 14
Binomial Distribution
Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.
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Slide 15
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Binomial Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
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Slide 16
Binomial Distribution
Binomial Probability Function
f (x ) =
n!
p x (1 − p )( n − x )
x !( n − x )!
where:
f(x) = the probability of x successes in n trials
n = the number of trials
p = the probability of success on any one trial
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Slide 17
Binomial Distribution
Binomial Probability Function
f (x) =
n!
p x (1 − p )( n − x )
x !(n − x )!
n!
x !( n − x )!
Number of experimental
outcomes providing exactly
x successes in n trials
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p x (1 − p)( n − x )
Probability of a particular
sequence of trial outcomes
with x successes in n trials
Slide 18
6
Binomial Distribution
Example: Evans Electronics
Evans is concerned about a low retention rate for
employees. In recent years, management has seen a
turnover of 10% of the hourly employees annually.
Thus, for any hourly employee chosen at random,
management estimates a probability of 0.1 that the
person will not be with the company next year.
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Slide 19
Binomial Distribution
Using the Binomial Probability Function
Choosing 3 hourly employees at random, what is
the probability that 1 of them will leave the company
this year?
Let: p = .10, n = 3, x = 1
n!
f ( x) =
p x (1 − p ) ( n − x )
x !( n − x )!
f (1) =
3!
(0.1)1 (0.9)2 = 3(.1)(.81) = .243
1!(3 − 1)!
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Slide 20
Binomial Distribution
Tree Diagram
1st Worker
2nd Worker
Leaves (.1)
Leaves
(.1)
3rd Worker
L (.1)
x
3
Prob.
.0010
S (.9)
2
.0090
L (.1)
2
.0090
S (.9)
1
.0810
Stays (.9)
Leaves (.1)
Stays
(.9)
L (.1)
2
.0090
S (.9)
1
.0810
1
.0810
0
.7290
L (.1)
Stays (.9)
S (.9)
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Slide 21
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Binomial Distribution
Using Tables of Binomial Probabilities
p
n
x
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
3
0
1
2
3
.8574
.1354
.0071
.0001
.7290
.2430
.0270
.0010
.6141
.3251
.0574
.0034
.5120
.3840
.0960
.0080
.4219
.4219
.1406
.0156
.3430
.4410
.1890
.0270
.2746
.4436
.2389
.0429
.2160
.4320
.2880
.0640
.1664
.4084
.3341
.0911
.1250
.3750
.3750
.1250
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Slide 22
Binomial Distribution
Expected Value
Variance
E(x) = µ = np
Var(x) = σ 2 = np(1 − p)
Standard Deviation
σ = np(1 − p )
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Slide 23
Binomial Distribution
Expected Value
E(x) = µ = 3(.1) = .3 employees out of 3
Variance
Var(x) = σ 2 = 3(.1)(.9) = .27
Standard Deviation
σ = 3(.1)(.9) = .52 employees
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Slide 24
8
Poisson Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
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Slide 25
Poisson Distribution
Examples of a Poisson distributed random variable:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a
toll booth in one hour
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Slide 26
Poisson Distribution
Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
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Slide 27
9
Poisson Distribution
Poisson Probability Function
f ( x) =
µ x e−µ
x!
where:
f(x) = probability of x occurrences in an interval
µ = mean number of occurrences in an interval
e = 2.71828
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Slide 28
Poisson Distribution
Example: Mercy Hospital
M ERCY
Patients arrive at the
emergency room of Mercy
Hospital at the average
rate of 6 per hour on
weekend evenings.
What is the
probability of 4 arrivals in
30 minutes on a weekend evening?
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Poisson Distribution
Slide 29
M ERCY
Using the Poisson Probability Function
µ = 6/hour = 3/half-hour, x = 4
f (4) =
34 (2.71828)−3
= .1680
4!
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Slide 30
10
Poisson Distribution
M ERCY
Using Poisson Probability Tables
µ
x
0
1
2
3
4
5
6
7
8
2.1
.1225
.2572
.2700
.1890
.0992
.0417
.0146
.0044
.0011
2.2
.1108
.2438
.2681
.1966
.1082
.0476
.0174
.0055
.0015
2.3
.1003
.2306
.2652
.2033
.1169
.0538
.0206
.0068
.0019
2.4
.0907
.2177
.2613
.2090
.1254
.0602
.0241
.0083
.0025
2.5
.0821
.2052
.2565
.2138
.1336
..0668
.0278
.0099
.0031
2.6
.0743
.1931
.2510
.2176
.1414
.0735
.0319
.0118
.0038
2.7
.0672
.1815
.2450
.2205
.1488
.0804
.0362
.0139
.0047
2.8
.0608
.1703
.2384
.2225
.1557
.0872
.0407
.0163
.0057
2.9
.0550
.1596
.2314
.2237
.1622
.0940
.0455
.0188
.0068
© 2005 Thomson/South-Western
Slide 31
Poisson Distribution
3.0
.0498
.1494
.2240
.2240
.1680
.1008
.0504
.0216
.0081
M ERCY
Poisson Distribution of Arrivals
Poisson Probabilities
Probability
0.25
0.20
actually,
the sequence
continues:
11, 12, …
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10
Number of Arrivals in 30 Minutes
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Slide 32
Poisson Distribution
A property of the Poisson distribution is that
the mean and variance are equal.
µ=σ2
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Slide 33
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Poisson Distribution
M ERCY
Variance for Number of Arrivals
During 30-Minute Periods
µ=σ2=3
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Slide 34
Hypergeometric Distribution
The hypergeometric distribution is closely related
to the binomial distribution.
However, for the hypergeometric distribution:
the trials are not independent, and
the probability of success changes from trial
to trial.
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Slide 35
Hypergeometric Distribution
Hypergeometric Probability Function
⎛ r ⎞⎛ N − r ⎞
⎟
⎜⎜ ⎟⎟⎜⎜
x n − x ⎟⎠
f ( x) = ⎝ ⎠⎝
⎛N⎞
⎜⎜ ⎟⎟
⎝n⎠
where:
for 0 < x < r
f(x) = probability of x successes in n trials
n = number of trials
N = number of elements in the population
r = number of elements in the population
labeled success
© 2005 Thomson/South-Western
Slide 36
12
Hypergeometric Distribution
Hypergeometric Probability Function
f (x) =
⎛r ⎞ ⎛N −r⎞
⎜x⎟ ⎜ n − x ⎟
⎝ ⎠ ⎝
⎠
⎛N⎞
⎜n⎟
⎝ ⎠
number of ways
x successes can be selected
from a total of r successes
in the population
for 0 < x < r
number of ways
n – x failures can be selected
from a total of N – r failures
in the population
number of ways
a sample of size n can be selected
from a population of size N
© 2005 Thomson/South-Western
Slide 37
Hypergeometric Distribution
Example: Neveready
Bob Neveready has removed two
dead batteries from a flashlight and
inadvertently mingled them with
the two good batteries he intended
as replacements. The four batteries look identical.
Bob now randomly selects two of the four
batteries. What is the probability he selects the two
good batteries?
ZAP
ZAP
ZAP
ZA
P
© 2005 Thomson/South-Western
Slide 38
Hypergeometric Distribution
Using the Hypergeometric Function
⎛ r ⎞⎛ N − r ⎞ ⎛ 2 ⎞⎛ 2 ⎞ ⎛ 2! ⎞⎛ 2! ⎞
⎜ x ⎟⎜ n − x ⎟ ⎜ 2 ⎟⎜ 0 ⎟ ⎜ 2!0! ⎟⎜ 0!2! ⎟
⎠ = ⎝ ⎠⎝ ⎠ = ⎝
⎠⎝
⎠ = 1 = .167
f ( x ) = ⎝ ⎠⎝
6
⎛N ⎞
⎛ 4⎞
⎛ 4! ⎞
⎜n⎟
⎜2⎟
⎜ 2!2! ⎟
⎝ ⎠
⎝ ⎠
⎝
⎠
where:
x = 2 = number of good batteries selected
n = 2 = number of batteries selected
N = 4 = number of batteries in total
r = 2 = number of good batteries in total
© 2005 Thomson/South-Western
Slide 39
13
Hypergeometric Distribution
Mean
⎛ r ⎞
E ( x) = µ = n ⎜ ⎟
⎝N⎠
Variance
r ⎞⎛ N − n ⎞
⎛ r ⎞⎛
Var ( x) = σ 22 = n ⎜ ⎟⎜ 1 − ⎟⎜
⎟
⎝ N ⎠⎝ N ⎠⎝ N − 1 ⎠
© 2005 Thomson/South-Western
Slide 40
Hypergeometric Distribution
Mean
⎛ r ⎞
⎛2⎞
⎟ = 2⎜ ⎟ = 1
⎝N⎠
⎝4⎠
µ = n ⎜⎛
Variance
⎛ 2 ⎞⎛ 2 ⎞⎛ 4 − 2 ⎞ 1
σ 22 = 2 ⎜ ⎟ ⎜1 − ⎟ ⎜
⎟ = = .333
⎝ 4 ⎠⎝
4 ⎠ ⎝ 4 −1 ⎠ 3
© 2005 Thomson/South-Western
Slide 41
Hypergeometric Distribution
Consider a hypergeometric distribution with n trials
and let p = (r/n) denote the probability of a success
on the first trial.
If the population size is large, the term (N – n)/(N – 1)
approaches 1.
The expected value and variance can be written
E(x) = np and Var(x) = np(1 – p).
Note that these are the expressions for the expected
value and variance of a binomial distribution.
continued
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Slide 42
14
Hypergeometric Distribution
When the population size is large, a hypergeometric
distribution can be approximated by a binomial
distribution with n trials and a probability of success
p = (r/N).
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Slide 43
End of Chapter 5
© 2005 Thomson/South-Western
Slide 44
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