Anderson
u
Sweeney
u
Williams
CONTEMPORARY
BUSINESS
STATISTICS
WITH MICROSOFT EXCEL
u Slides Prepared by JOHN LOUCKS u
© 2001 South-Western/Thomson Learning
Slide 1
Chapter 5
Discrete Probability Distributions






Random Variables
Discrete Probability
Distributions
Expected Value and
Variance
Binomial Probability
Distribution
Poisson Probability
Distribution
Hypergeometric Probability
Distribution
Slide 2
Random Variables




A random variable is a numerical description of the
outcome of an experiment.
A random variable can be classified as being either
discrete or continuous depending on the numerical
values it assumes.
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.
Slide 3
Example: JSL Appliances

Discrete random variable with a finite number of
values
Let x = number of TV sets sold at the store in one day
where x can take on 5 values (0, 1, 2, 3, 4)

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.
Slide 4
Discrete Probability Distributions




The probability distribution for a random variable
describes how probabilities are distributed over the
values of the random variable.
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
f(x) = 1
We can describe a discrete probability distribution
with a table, graph, or equation.
Slide 5
Example: JSL Appliances

Using past data on TV sales (below left), a tabular
representation of the probability distribution for TV
sales (below right) was developed.
Units Sold
0
1
2
3
4
Number
of Days
80
50
40
10
20
200
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
1.00
Slide 6
Example: JSL Appliances
Graphical Representation of the Probability
Distribution
.50
Probability

.40
.30
.20
.10
0
1
2
3
4
Values of Random Variable x (TV sales)
Slide 7
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
where:

n = the number of values the random
variable may assume
Note that the values of the random variable are
equally likely.
Slide 8
Expected Value and Variance



The expected value, or mean, of a random variable is
a measure of its central location.
• Expected value of a discrete random variable:
E(x) =  = xf(x)
The variance summarizes the variability in the values
of a random variable.
• Variance of a discrete random variable:
Var(x) =  2 = (x - )2f(x)
The standard deviation, , is defined as the positive
square root of the variance.
Slide 9
Example: JSL Appliances

Expected Value of a Discrete Random Variable
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
xf(x)
.00
.25
.40
.15
.40
1.20 = E(x)
The expected number of TV sets sold in a day is 1.2
Slide 10
Example: JSL Appliances

Variance and Standard Deviation
of a Discrete Random Variable
x
x-
(x - )2
f(x)
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
(x - )2f(x)
.576
.010
.128
.162
.784
1.660 =  
The variance of daily sales is 1.66 TV sets squared.
The standard deviation of sales is 1.2884 TV sets.
Slide 11
Using Excel to Compute the Expected Value,
Variance, and Standard Deviation

1
2
3
4
5
6
7
8
9
10
Formula Worksheet
A
Sales
0
1
2
3
4
B
Probability
0.40
0.25
0.20
0.05
0.10
C
Sq.Dev.from Mean
=(A2-$B$8)^2
=(A3-$B$8)^2
=(A4-$B$8)^2
=(A5-$B$8)^2
=(A6-$B$8)^2
Mean =SUMPRODUCT(A2:A6,B2:B6)
Variance =SUMPRODUCT(C2:C6,B2:B6)
Std.Dev. =SQRT(B9)
Slide 12
Using Excel to Compute the Expected Value,
Variance, and Standard Deviation

1
2
3
4
5
6
7
8
9
10
Value Worksheet
A
Sales
0
1
2
3
4
B
Probability
0.40
0.25
0.20
0.05
0.10
C
Sq.Dev.from Mean
1.44
0.04
0.64
3.24
7.84
Mean 1.2
Variance 1.66
Std.Dev. 1.2884
Slide 13
Binomial Probability Distribution

Properties of a Binomial Experiment
• The experiment consists of a sequence of n
identical trials.
• Two outcomes, success and failure, are possible on
each trial.
• The probability of a success, denoted by p, does
not change from trial to trial.
• The trials are independent.
Slide 14
Example: Evans Electronics

Binomial Probability Distribution
Evans is concerned about a low retention rate for
employees. On the basis of past experience,
management has seen a turnover of 10% of the
hourly employees annually. Thus, for any hourly
employees chosen at random, management estimates
a probability of 0.1 that the person will not be with
the company next year.
Choosing 3 hourly employees a random, what is
the probability that 1 of them will leave the company
this year?
Let:
p = .10, n = 3, x = 1
Slide 15
Binomial Probability Distribution

Binomial Probability Function
n!
f ( x) 
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
Slide 16
Example: Evans Electronics

Using the Binomial Probability Function
n!
f ( x) 
p x (1  p ) (n  x )
x !( n  x )!
3!
f (1) 
( 0.1)1 ( 0. 9 ) 2
1!( 3  1)!
= (3)(0.1)(0.81)
= .243
Slide 17
Example: Evans Electronics

n
3
Using the Tables of Binomial Probabilities
x
0
1
2
3
.10
.7290
.2430
.0270
.0010
.15
.6141
.3251
.0574
.0034
.20
.5120
.3840
.0960
.0080
.25
.4219
.4219
.1406
.0156
p
.30
.3430
.4410
.1890
.0270
.35
.2746
.4436
.2389
.0429
.40
.2160
.4320
.2880
.0640
.45
.1664
.4084
.3341
.0911
.50
.1250
.3750
.3750
.1250
Slide 18
Example: Evans Electronics

Using a Tree Diagram
First
Worker
Second
Worker
Leaves (.1)
Leaves (.1)
Third
Worker
3
.0010
S (.9)
2
.0090
2
.0090
1
.0810
L (.1)
2
.0090
S (.9)
1
.0810
L (.1)
1
.0810
S (.9)
0
.7290
L (.1)
S (.9)
Stays (.9)
Stays (.9)
Probab.
L (.1)
Stays (.9)
Leaves (.1)
Value
of x
Slide 19
Using Excel to Compute
Binomial Probabilities

Formula Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
=BINOMDIST(A5,$A$1,$A$2,FALSE)
=BINOMDIST(A6,$A$1,$A$2,FALSE)
=BINOMDIST(A7,$A$1,$A$2,FALSE)
=BINOMDIST(A8,$A$1,$A$2,FALSE)
Slide 20
Using Excel to Compute
Binomial Probabilities

Value Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
0.729
0.243
0.027
0.001
Slide 21
Using Excel to Compute
Cumulative Binomial Probabilities

Formula Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
=BINOMDIST(A5,$A$1,$A$2,TRUE)
=BINOMDIST(A6,$A$1,$A$2,TRUE)
=BINOMDIST(A7,$A$1,$A$2,TRUE)
=BINOMDIST(A8,$A$1,$A$2,TRUE)
Slide 22
Using Excel to Compute
Cumulative Binomial Probabilities

Value Worksheet
A
1
2
3
4
5
6
7
8
9
x
0
1
2
3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
0.729
0.972
0.999
1.000
Slide 23
Binomial Probability Distribution

Expected Value
E(x) =  = np

Variance

Var(x) =  2 = np(1 - p)
Standard Deviation
SD( x )    np(1  p)
Slide 24
Example: Evans Electronics

Binomial Probability Distribution
• Expected Value
E(x) =  = 3(.1) = .3 employees out of 3
• Variance
Var(x) =  2 = 3(.1)(.9) = .27
• Standard Deviation
SD( x)    3(.1)(.9)  .52 employees
Slide 25
Poisson Probability Distribution

Properties of a Poisson Experiment
• The probability of an occurrence is the same for
any two intervals of equal length.
• The occurrence or nonoccurrence in any interval is
independent of the occurrence or nonoccurrence
in any other interval.
Slide 26
Poisson Probability Distribution

Poisson Probability Function
f ( x) 
 x e 
x!
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
Slide 27
Example: Mercy Hospital

Using the Poisson Probability Function
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?
 = 6/hour = 3/half-hour, x = 4
34 ( 2. 71828) 3
f ( 4) 
.1680
4!
Slide 28
Example: Mercy Hospital

Using the Tables of Poisson Probabilities

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
3.0
.0498
.1494
.2240
.2240
.1680
.1008
.0504
.0216
.0081
Slide 29
Using Excel to Compute
Poisson Probabilities

Formula Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
Etc.
Probability f (x )
=POISSON(A4,$A$1,FALSE)
=POISSON(A5,$A$1,FALSE)
=POISSON(A6,$A$1,FALSE)
=POISSON(A7,$A$1,FALSE)
=POISSON(A8,$A$1,FALSE)
=POISSON(A9,$A$1,FALSE)
=POISSON(A10,$A$1,FALSE)
Etc.
Slide 30
Using Excel to Compute
Poisson Probabilities

Value Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
Etc.
Probability f (x )
0.0498
0.1494
0.2240
0.2240
0.1680
0.1008
0.0504
Etc.
Slide 31
Using Excel to Compute
Cumulative Poisson Probabilities

Formula Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
Etc.
Cumulative Probability
=POISSON(A4,$A$1,TRUE)
=POISSON(A5,$A$1,TRUE)
=POISSON(A6,$A$1,TRUE)
=POISSON(A7,$A$1,TRUE)
=POISSON(A8,$A$1,TRUE)
=POISSON(A9,$A$1,TRUE)
=POISSON(A10,$A$1,TRUE)
Etc.
Slide 32
Using Excel to Compute
Cumulative Poisson Probabilities

Value Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
Etc.
Cumulative Probability
0.0498
0.1991
0.4232
0.6472
0.8153
0.9161
0.9665
Etc.
Slide 33
Hypergeometric Probability Distribution


The hypergeometric distribution is closely related to
the binomial distribution.
With the hypergeometric distribution, the trials are
not independent, and the probability of success
changes from trial to trial.
Slide 34
Hypergeometric Probability Distribution

Hypergeometric Probability Function
where:
 r  N  r 
 

x  n  x 

f ( x) 
N
 
n
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
Slide 35
Example: Neveready

Hypergeometric Probability Distribution
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?
Slide 36
Example: Neveready

Hypergeometric Probability Distribution
 r  N  r   2  2   2!  2! 
 
      
x  n  x   2  0   2!0! 0!2! 1

f ( x) 


  .167
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
Slide 37
Using Excel to Compute
Hypergeometric Probabilities

Formula Worksheet
A
1
2
2
2
3
2
4
4
5
6 f (x ) =
7
B
=
=
=
=
Number
Number
Number
Number
of Successes (x )
of Trials (n )
of Elements in the Population Labeled Success (r )
of Elements in the Population (N )
=HYPGEOMDIST(A1,A2,A3,A4)
Slide 38
Using Excel to Compute
Hypergeometric Probabilities

Value Worksheet
A
1
2
2
2
3
2
4
4
5
6 f (x ) =
7
B
=
=
=
=
Number
Number
Number
Number
of Successes (x )
of Trials (n )
of Elements in the Population Labeled Success (r )
of Elements in the Population (N )
0.167
Slide 39
End of Chapter 5
Slide 40