Download Chap. 4: Discrete Random Variables

© 1998 Prentice-Hall, Inc.
Statistics for
Business & Economics
Discrete Random Variables
Chapter 4
4-1
Learning Objectives
© 1998 Prentice-Hall, Inc.
1. Define random variable
2. Compute the expected value & variance
of discrete random variables
3. Describe the binomial & Poisson
probability distributions
4. Calculate probabilities for binomial &
Poisson random variables
4-2
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re taking a 33
question multiple choice
test. Each question has
4 choices. Clueless on 1
question, you decide to
guess. What’s the
chance you’ll get it right?
If you guessed on all 33
questions, what would be
your grade? Pass?
4-3
Alone
Group Class
Random Variable
© 1998 Prentice-Hall, Inc.
1. A numerical outcome of an experiment
2. May be discrete or continuous
3. Discrete random variable


Countable number of values
Example: Number of tails in 2 coin tosses
4. Continuous random variable


Infinite number of values within an interval
Example: Amount of soda in a 12 oz. can
4-4
© 1998 Prentice-Hall, Inc.
Discrete Random Variables
4-5
Discrete
Random Variable
© 1998 Prentice-Hall, Inc.
1.
2.
3.
4.
Type of random variable
Whole number (0, 1, 2, 3 etc.)
Obtained by counting
Usually finite number of values

Poisson random variable is exception ()
4-6
© 1998 Prentice-Hall, Inc.
4-7
Discrete Random
Variable Examples
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
4-8
Random
Variable
Possible
Values
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Make 100 sales calls
4-9
Random
Variable
Possible
Values
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
4 - 10
Possible
Values
0, 1, 2, ..., 100
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
4 - 11
Possible
Values
0, 1, 2, ..., 100
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
4 - 12
Possible
Values
0, 1, 2, ..., 100
# Defective 0, 1, 2, ..., 70
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
Answer 33 questions
4 - 13
Possible
Values
0, 1, 2, ..., 100
# Defective 0, 1, 2, ..., 70
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
0, 1, 2, ..., 100
# Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct
4 - 14
Possible
Values
0, 1, 2, ..., 33
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
4 - 15
0, 1, 2, ..., 100
# Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct
Count cars at toll
between 11:00 & 1:00
Possible
Values
0, 1, 2, ..., 33
© 1998 Prentice-Hall, Inc.
Discrete Random
Variable Examples
Experiment
Random
Variable
Make 100 sales calls # Sales
Inspect 70 radios
Possible
Values
0, 1, 2, ..., 100
# Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct
0, 1, 2, ..., 33
Count cars at toll
# Cars
between 11:00 & 1:00 arriving
0, 1, 2, ..., 
4 - 16
Discrete
Probability Distribution
© 1998 Prentice-Hall, Inc.
1. List of all possible [x, p(x)] pairs


x = Value of random variable (outcome)
p(x) = Probability associated with value
2. Mutually exclusive (no overlap)
3. Collectively exhaustive (nothing left out)
4. 0  p(x)  1 (or p(x)  0)
5.  p(x) = 1
4 - 17
© 1998 Prentice-Hall, Inc.
4 - 18
Discrete Probability
Distribution Example
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
4 - 19
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
4 - 20
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
© 1984-1994 T/Maker Co.
4 - 21
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
© 1984-1994 T/Maker Co.
4 - 22
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
1
© 1984-1994 T/Maker Co.
4 - 23
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
1
© 1984-1994 T/Maker Co.
4 - 24
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
1
2
© 1984-1994 T/Maker Co.
4 - 25
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
1
2
© 1984-1994 T/Maker Co.
4 - 26
1/4 = .25
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
0
1/4 = .25
1
2/4 = .50
2
© 1984-1994 T/Maker Co.
4 - 27
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, x Probabilities, p(x)
© 1984-1994 T/Maker Co.
4 - 28
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
© 1998 Prentice-Hall, Inc.
Visualizing Discrete
Probability Distributions
4 - 29
© 1998 Prentice-Hall, Inc.
Visualizing Discrete
Probability Distributions
Listing
{ (0, .25), (1, .50), (2, .25) }
4 - 30
© 1998 Prentice-Hall, Inc.
Visualizing Discrete
Probability Distributions
Listing
{ (0, .25), (1, .50), (2, .25) }
4 - 31
Table
# Tails
f(x)
Count
p(x)
0
1
2
1
2
1
.25
.50
.25
© 1998 Prentice-Hall, Inc.
Visualizing Discrete
Probability Distributions
Listing
Table
{ (0, .25), (1, .50), (2, .25) }
p(x)
.50
.25
.00
Graph
x
0
4 - 32
1
2
# Tails
f(x)
Count
p(x)
0
1
2
1
2
1
.25
.50
.25
© 1998 Prentice-Hall, Inc.
Visualizing Discrete
Probability Distributions
Listing
Table
{ (0, .25), (1, .50), (2, .25) }
p(x)
.50
.25
.00
Graph
4 - 33
1
f(x)
Count
p(x)
0
1
2
1
2
1
.25
.50
.25
Equation
x
0
# Tails
2
n!
p ( x) 
p x (1  p) n  x
x !(n  x)!
Summary Measures
© 1998 Prentice-Hall, Inc.
1. Expected value



Mean of probability distribution
Weighted average of all possible values
 = E(X) = x p(x)
2. Variance


Weighted average squared deviation
about mean
2 = E[ (x (x  p(x)
4 - 34
© 1998 Prentice-Hall, Inc.
x
p(x)
Total
4 - 35
Summary Measures
Calculation Table
x p(x )
x p(x )
x-
(x -)
2
(x -) p( x )
2
 (x -) p( x )
2
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You toss 2 coins. You’re
interested in the number
of tails. What are the
expected value &
standard deviation of
this random variable,
number of tails?
© 1984-1994 T/Maker Co.
4 - 36
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Expected Value &
Variance Solution*
2
x
p(x)
x p(x )
x-
(x -)
0
.25
0
-1.00
1.00
.25
1
.50
.50
0
0
0
2
.25
.50
1.00
1.00
.25
 = 1.0
4 - 37
(x -) p( x )
2
 = .50
2
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Function
4 - 38
Discrete Probability
Distribution Function
© 1998 Prentice-Hall, Inc.
1.
Type of model

2.
Representation of some
underlying phenomenon
Mathematical formula
3.
Represents discrete
random variable
4.
Used to get exact
probabilities
4 - 39
P (X  x )
 e
x
x!
-

© 1998 Prentice-Hall, Inc.
4 - 40
Discrete Probability
Distribution Models
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
4 - 41
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
Binomial
4 - 42
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
Binomial
4 - 43
Poisson
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
Binomial
4 - 44
Poisson
Other
© 1998 Prentice-Hall, Inc.
Binomial Distribution
4 - 45
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
Binomial
4 - 46
Poisson
Other
© 1998 Prentice-Hall, Inc.
Binomial
Random Variable
1. Number of ‘successes’ in a sample of n
observations (trials)
4 - 47
Binomial
Random Variable
© 1998 Prentice-Hall, Inc.
1. Number of ‘successes’ in a sample of n
observations (trials)
2. Examples

# Reds in 15 spins of roulette wheel
# Defective items in a batch of 5 items

# Correct on a 33 question exam

# Customers who purchase out of 100
customers who enter store

4 - 48
Binomial Distribution
Characteristics
© 1998 Prentice-Hall, Inc.
1. Sequence of n identical trials
2. Each trial has 2 outcomes

‘Success’ (desired outcome) or ‘failure’
3. Constant trial probability
4. Trials are independent
5. Two different sampling methods


Infinite population with replacement
Finite population without replacement
4 - 49
© 1998 Prentice-Hall, Inc.
Binomial Probability
Distribution Function
nI
F
p( x )  GJp q
Hx K
x
n x
n!
x
n x

p (1  p)
x !(n  x )!
p(x) = Probability of x ‘successes’ in n trials
n = Sample size
p = Probability of ‘success’
x = Number of ‘successes’ in sample
(x = 0, 1, 2, ..., n)
4 - 50
© 1998 Prentice-Hall, Inc.
4 - 51
Binomial Distribution
Characteristics
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Characteristics
Mean
  E ( x )  np
  np (1  p)
4 - 52
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Characteristics
Mean
  E ( x )  np
Standard Deviation
  np (1  p)
4 - 53
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Characteristics
Mean
  E ( x )  np
P(X)
.6
.4
.2
.0
  np (1  p)
X
0
Standard Deviation
P(X)
.6
.4
.2
.0
1
2
3
4
5
n = 5 p = 0.5
X
0
4 - 54
n = 5 p = 0.1
1
2
3
4
5
© 1998 Prentice-Hall, Inc.
Binomial Probability
Distribution Example
Experiment: Toss 1 coin 5 times in a row. Note
# tails. What’s the probability of 3 tails?
4 - 55
Binomial Probability
Distribution Example
© 1998 Prentice-Hall, Inc.
Experiment: Toss 1 coin 5 times in a row. Note
# tails. What’s the probability of 3 tails?
n!
x
n x
p( x ) 
p (1  p )
x !(n  x )!
5!
3
5 3
p(3) 
.5 (1 .5)
3 !(5  3)!
 0.3125
4 - 56
© 1998 Prentice-Hall, Inc.
4 - 57
Using the Binomial
Probability Table
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probabilities: p(x  k) given n & p
4 - 58
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probabilities: p(x  k) given n & p
4 - 59
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probabilities: p(x  k) given n & p
4 - 60
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probabilities: p(x  k) given n & p
4 - 61
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Select table for n = 5
4 - 62
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Select row for k = 3
4 - 63
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Select column for p = 0.50
4 - 64
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probability: p(x  3) = .812
4 - 65
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
p(x  3) = p(x  3) - p(x  2). Select row for k = 2
4 - 66
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Select column for p = 0.50
4 - 67
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
Cumulative probability: p(x  2) = .500
4 - 68
© 1998 Prentice-Hall, Inc.
Using the Binomial
Probability Table
n=5
p
k
.01
…
0.50
…
.99
0
.951
…
.031
…
.000
1
.999
…
.188
…
.000
2
1.000
…
.500
…
.000
3
1.000
…
.812
…
.001
4
1.000
…
.969
…
.049
p(x  3) = p(x  3) - p(x  2) = .812 - .500 = .312
4 - 69
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Thinking Challenge
You’re a telemarketer selling
service contracts for Macy’s.
You’ve sold 20 in your last
100 calls (p = .20). If you
call 12 people tonight,
what’s the probability of
A.
B.
C.
D.
No sales?
Exactly 2 sales?
At most 2 sales?
At least 2 sales?
4 - 70
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Solution*
Using the Binomial Formula:
A. p(0) = .0687
B. p(2) = .2835
C. p(at most 2) = p(0) + p(1) + p(2)
= .0687 + .2062 + .2835
= .5584
D. p(at least 2) = p(2) + p(3)...+ p(12)
= 1 - [p(0) + p(1)]
= 1 - .0687 - .2062
= .7251
4 - 71
© 1998 Prentice-Hall, Inc.
Poisson Distribution
4 - 72
© 1998 Prentice-Hall, Inc.
Discrete Probability
Distribution Models
Discrete
Probability
Distribution
Binomial
4 - 73
Poisson
Other
Poisson Random Variable
© 1998 Prentice-Hall, Inc.
1. Number of events that occur in an
interval

Events per unit

Time, length, area, space
2. Examples



# Customers arriving in 20 minutes
# Strikes per year in the U.S.
# Defects per lot (group) of VCR’s
4 - 74
Poisson Process
© 1998 Prentice-Hall, Inc.
1. Constant event
probability

2.
Average of 60/hr is 1/min
for 60 1-minute intervals
One event per interval

3.
Don’t arrive together
Independent events

Arrival of 1 person does
not affect another’s arrival
4 - 75
© 1984-1994 T/Maker Co.
© 1998 Prentice-Hall, Inc.
Poisson Probability
Distribution Function
x -
p ( x) 
e
x!
p(x) = Probability of x given 
 = Expected (mean) number of ‘successes’
e = 2.71828 (base of natural logs)
x = Number of ‘successes’ per unit
4 - 76
© 1998 Prentice-Hall, Inc.
4 - 77
Poisson Distribution
Characteristics
Poisson Distribution
Characteristics
© 1998 Prentice-Hall, Inc.
Mean
  E(x)  

N
 x p( x )
i 1
 
4 - 78
Poisson Distribution
Characteristics
© 1998 Prentice-Hall, Inc.
Mean
  E(x)  

N
 x p( x )
i 1
Standard Deviation
 
4 - 79
Poisson Distribution
Characteristics
© 1998 Prentice-Hall, Inc.
Mean
  E(x)  

= 0.5
P(X)
N
 x p( x )
.6
.4
.2
.0
X
0
1
2
3
4
5
i 1
= 6
P(X)
Standard Deviation
 
.6
.4
.2
.0
X
0
4 - 80
2
4
6
8
10
© 1998 Prentice-Hall, Inc.
Poisson Distribution
Example
Patients arrive at a
hospital clinic at a rate
of 72 per hour. What
is the probability of
4 patients arriving in
3 minutes?
4 - 81
© 1995 Corel Corp.
© 1998 Prentice-Hall, Inc.
Poisson Distribution
Solution
72 per hr. = 1.2 per min. = 3.6 per 3 min. interval
4 - 82
© 1998 Prentice-Hall, Inc.
Poisson Distribution
Solution
72 per hr. = 1.2 per min. = 3.6 per 3 min. interval
p( x ) 
 e
x
-
x!
3.6f e
a
p(4) 
4
4!
 0.1912
4 - 83
-3.6
© 1998 Prentice-Hall, Inc.
4 - 84
Using the Poisson
Probability Table
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
Cumulative probabilities
4 - 85
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
Select row with  = 3.6
4 - 86
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
:
:
.558 .744
.515 .706
.473 .668
:
:
p(x  4) = p(x  4) - p(x  3).
4 - 87
4
…
9
:
…
…
…
:
:
.997
.996
.994
:
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
p(x  4) = p(x  4) - p(x  3). Select column x = 4.
4 - 88
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
p(x  4) = p(x  4) - p(x  3) = .706 - p(x  3)
4 - 89
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
Select column x = 3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
p(x  4) = p(x  4) - p(x  3) = .706 - p(x  3)
4 - 90
Using the Poisson
Probability Table
© 1998 Prentice-Hall, Inc.

.02
:
3.4
3.6
3.8
:
0
.980
:
.033
.027
.022
:
…
…
:
…
…
…
:
x
3
4
:
:
.558 .744
.515 .706
.473 .668
:
:
…
9
:
…
…
…
:
:
.997
.996
.994
:
p(x  4) = p(x  4) - p(x  3) = .706 - .515 = .191
4 - 91
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You work in Quality
Assurance for an
investment firm. A
clerk enters 75 words
per minute with 6
errors per hour. What
is the probability of 0
errors in a 255-word
bond transaction?
© 1984-1994 T/Maker Co.
4 - 92
Alone
Group Class
© 1998 Prentice-Hall, Inc.
Poisson Distribution
Solution: Finding *
75 words/min = (75 words/min)(60 min/hr)
= 4500 words/hr
6 errors/hr = 6 errors/4500 words
= .00133 errors/word
In a 255-word transaction (interval):
 = (.00133 errors/word )(255 words)
= .34 errors/255-word transaction
4 - 93
© 1998 Prentice-Hall, Inc.
Poisson Distribution
Solution: Finding p(0)*
 e
x
p (x) 
-
x!
.34f e
a
p (0 ) 
0 -.34
0!
= .7118
4 - 94
Conclusion
© 1998 Prentice-Hall, Inc.
1. Defined random variable
2. Computed the expected value &
variance of discrete random variables
3. Described the binomial & Poisson
probability distributions
4. Calculated probabilities for binomial &
Poisson random variables
4 - 95
This Class...
© 1998 Prentice-Hall, Inc.
Please take a moment to answer the
following questions in writing:
1. What was the most important thing you
learned in class today?
2. What do you still have questions about?
3. How can today’s class be improved?
4 - 96
End of Chapter
Any blank slides that follow are
blank intentionally.