Download Business Statistics: A First Course, 3rd Edition

Business Statistics:
A First Course
(3rd Edition)
Chapter 5
Probability Distributions
© 2003 Prentice-Hall, Inc.
Chap 5-1
Chapter Topics

The Probability of a Discrete Random Variable

Covariance and its Applications in Finance
Binomial Distribution

The Normal Distribution

The Standardized Normal Distribution

Evaluating the Normality Assumption

© 2003 Prentice-Hall, Inc.
Chap 5-2
Random Variable

Random Variable


Outcomes of an experiment expressed numerically
e.g. Toss a die twice; count the number of times
the number 4 appears (0, 1 or 2 times)
© 2003 Prentice-Hall, Inc.
Chap 5-3
Discrete Random Variable

Discrete Random Variable

Obtained by Counting (0, 1, 2, 3, etc.)

Usually a finite number of different values

e.g. Toss a coin 5 times; count the number of tails
(0, 1, 2, 3, 4, or 5 times)
© 2003 Prentice-Hall, Inc.
Chap 5-4
Discrete Probability Distribution
Example
Event: Toss 2 Coins.
Count # Tails.
Probability Distribution
Values
Probability
T
T
T
© 2003 Prentice-Hall, Inc.
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
T
Chap 5-5
Discrete Probability Distribution

List of All Possible [Xj , P(Xj) ] Pairs

Xj = Value of random variable

P(Xj) = Probability associated with value

Mutually Exclusive (Nothing in Common)

Collective Exhaustive (Nothing Left Out)
0  P X j  1
© 2003 Prentice-Hall, Inc.
P X  1
j
Chap 5-6
Summary Measures

Expected value (The Mean)

Weighted average of the probability distribution
  E X    X jP X j 

j

e.g. Toss 2 coins, count the number of tails,
compute expected value
   X jP X j 
j
© 2003 Prentice-Hall, Inc.
  0 .25  1.5   2 .25  1
Chap 5-7
Summary Measures

(continued)
Variance



Weighted average squared deviation about the mean
  E  X        X j    P  X j 
2
2
2


e.g. Toss 2 coins, count number of tails, compute
variance
   X j    P X j 
2
2
  0  1 .25   1  1 .5    2  1 .25   .5
2
© 2003 Prentice-Hall, Inc.
2
2
Chap 5-8
Covariance and its Application
N
 XY    X i  E  X   Yi  E Y  P  X iYi 
i 1
X : discrete random variable
X i : i th outcome of X
Y : discrete random variable
Yi : i th outcome of Y
P  X iYi  : probability of occurrence of the i
th
outcome of X and the i th outcome of Y
© 2003 Prentice-Hall, Inc.
Chap 5-9
Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments
P(Xi) P(Yi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
.2
Recession
-$100
-$200
.5
.5
Stable Economy
+ 100
+ 50
.3
.3
Expanding Economy
+ 250
+ 350
E  X    X   100.2  100.5   250.3  $105
E Y   Y   200.2   50.5   350.3  $90
© 2003 Prentice-Hall, Inc.
Chap 5-10
Computing the Variance for
Investment Returns
P(Xi) P(Yi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
.2
Recession
-$100
-$200
.5
.5
Stable Economy
+ 100
+ 50
.3
.3
Expanding Economy
+ 250
+ 350
  .2  100  105   .5 100  105   .3 250  105 
2
2
X
2
 X  121.35
 14, 725
  .2  200  90   .5  50  90   .3 350  90 
2
2
Y
 37,900
© 2003 Prentice-Hall, Inc.
2
2
2
 Y  194.68
Chap 5-11
Computing the Covariance for
Investment Returns
P(XiYi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
 XY   100  105 200  90 .2   100  105 50  90 .5
  250  105  350  90 .3  23,300
The Covariance of 23,000 indicates that the two investments are
positively related and will vary together in the same direction.
© 2003 Prentice-Hall, Inc.
Chap 5-12
Important Discrete Probability
Distributions
Discrete Probability
Distributions
Binomial
© 2003 Prentice-Hall, Inc.
Chap 5-13
Binomial Probability Distribution

‘n’ Identical Trials


2 Mutually Exclusive Outcomes on Each Trial


e.g. 15 tosses of a coin; 10 light bulbs taken from
a warehouse
e.g. Head or tail in each toss of a coin; defective or
not defective light bulb
Trials are Independent

The outcome of one trial does not affect the
outcome of the other
© 2003 Prentice-Hall, Inc.
Chap 5-14
Binomial Probability Distribution
(continued)

Constant Probability for Each Trial


e.g. Probability of getting a tail is the same each
time we toss the coin
2 Sampling Methods


Infinite population without replacement
Finite population with replacement
© 2003 Prentice-Hall, Inc.
Chap 5-15
Binomial Probability Distribution
Function
n!
n X
X
P X  
p 1  p 
X ! n  X  !
P  X  : probability of X successes given n and p
X : number of "successes" in sample  X  0,1,
, n
p : the probability of each "success"
Tails in 2 Tosses of Coin
n : sample size
© 2003 Prentice-Hall, Inc.
X
0
P(X)
1/4 = .25
1
2/4 = .50
2
1/4 = .25
Chap 5-16
Binomial Distribution
Characteristics

Mean
  E  X   np
 E.g.   np  5 .1  .5


Variance and
Standard Deviation

 2  np 1  p 
  np 1  p 

P(X)
.6
.4
.2
0
n = 5 p = 0.1
X
0
1
2
3
4
5
e.g.
  np 1  p   5 .11  .1  .6708
© 2003 Prentice-Hall, Inc.
Chap 5-17
Binomial Distribution in PHStat


PHStat | Probability & Prob. Distributions |
Binomial
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 5-18
Continuous Probability
Distributions

Continuous Random Variable



Continuous Probability Distribution


Values from interval of numbers
Absence of gaps
Distribution of continuous random variable
Most Important Continuous Probability
Distribution

The normal distribution
© 2003 Prentice-Hall, Inc.
Chap 5-19
The Normal Distribution





“Bell Shaped”
Symmetrical
Mean, Median and
Mode are Equal
Interquartile Range
Equals 1.33 
Random Variable
has Infinite Range
© 2003 Prentice-Hall, Inc.
f(X)

X
Mean
Median
Mode
Chap 5-20
The Mathematical Model
2
1
 (1/ 2)  X    /  
f X  
e
2
f  X  : density of random variable X
  3.14159;
e  2.71828
 : population mean
 : population standard deviation
X : value of random variable    X   
© 2003 Prentice-Hall, Inc.
Chap 5-21
Many Normal Distributions
There are an Infinite Number of Normal Distributions
Varying the Parameters  and , we obtain
Different Normal Distributions
© 2003 Prentice-Hall, Inc.
Chap 5-22
Finding Probabilities
Probability is
the area under
the curve!
P c  X  d   ?
f(X)
c
© 2003 Prentice-Hall, Inc.
d
X
Chap 5-23
Which Table to Use?
Infinitely Many Normal Distributions Mean
Infinitely Many Tables to Look Up!
© 2003 Prentice-Hall, Inc.
Chap 5-24
Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
Z 1
.02
.5478
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Probabilities
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.12
Only One Table is Needed
Chap 5-25
Standardizing Example
Z
X 

Standardized
Normal Distribution
Normal Distribution
  10
Z 1
6.2
© 2003 Prentice-Hall, Inc.
6.2  5

 0.12
10
 5
X
0.12
Z  0
Z
Chap 5-26
Example:
P  2.9  X  7.1  .1664
Z
X 

2.9  5

 .21
10
Z
X 

7.1  5

 .21
10
Standardized
Normal Distribution
Normal Distribution
  10
.0832
Z 1
.0832
2.9 7.1
 5
© 2003 Prentice-Hall, Inc.
X
0.21 0.21
Z  0
Z
Chap 5-27
Example:
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
Z 1
.5832
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.21
Chap 5-28
Example:
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z  0
Z 1
.4168
-0.3 .3821 .3783 .3745
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
© 2003 Prentice-Hall, Inc.
0
Z = -0.21
Chap 5-29
Normal Distribution in PHStat


PHStat | Probability & Prob. Distributions |
Normal …
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 5-30
Example:
P  X  8  .3821
Z
X 

85

 .30
10
Standardized
Normal Distribution
Normal Distribution
  10
Z 1
.3821
 5
© 2003 Prentice-Hall, Inc.
8
X
0.30
Z  0
Z
Chap 5-31
Example:
P  X  8  .3821
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
(continued)
Z 1
.6179
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.30
Chap 5-32
Finding Z Values for Known
Probabilities
What is Z Given
Probability = 0.6217 ?
Z  0
Z 1
Cumulative Standardized
Normal Distribution Table
(Portion)
Z
.00
.01
0.2
0.0 .5000 .5040 .5080
.6217
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0
Z  .31
© 2003 Prentice-Hall, Inc.
0.3 .6179 .6217 .6255
Chap 5-33
Recovering X Values for Known
Probabilities
Standardized
Normal Distribution
Normal Distribution
  10
.6179
Z 1
.3821
 5
?
X
Z  0
0.30
X    Z  5  .3010  8
© 2003 Prentice-Hall, Inc.
Z
Chap 5-34
Assessing Normality


Not All Continuous Random Variables are
Normally Distributed
It is Important to Evaluate how Well the Data
Set Seems to be Adequately Approximated by
a Normal Distribution
© 2003 Prentice-Hall, Inc.
Chap 5-35
Assessing Normality

Construct Charts



(continued)
For small- or moderate-sized data sets, do stemand-leaf display and box-and-whisker plot look
symmetric?
For large data sets, does the histogram or polygon
appear bell-shaped?
Compute Descriptive Summary Measures



Do the mean, median and mode have similar
values?
Is the interquartile range approximately 1.33 ?
Is the range approximately 6 ?
© 2003 Prentice-Hall, Inc.
Chap 5-36
Assessing Normality

Observe the Distribution of the Data Set




(continued)
Do approximately
between mean 
Do approximately
between mean 
Do approximately
between mean 
2/3 of the observations lie
1 standard deviation?
4/5 of the observations lie
1.28 standard deviations?
19/20 of the observations lie
2 standard deviations?
Evaluate Normal Probability Plot

Do the points lie on or close to a straight line with
positive slope?
© 2003 Prentice-Hall, Inc.
Chap 5-37
Assessing Normality

(continued)
Normal Probability Plot




Arrange Data into Ordered Array
Find Corresponding Standardized Normal Quantile
Values
Plot the Pairs of Points with Observed Data Values
on the Vertical Axis and the Standardized Normal
Quantile Values on the Horizontal Axis
Evaluate the Plot for Evidence of Linearity
© 2003 Prentice-Hall, Inc.
Chap 5-38
Assessing Normality
(continued)
Normal Probability Plot for Normal
Distribution
90
X 60
Z
30
-2 -1 0 1 2
Look for a Straight Line!
© 2003 Prentice-Hall, Inc.
Chap 5-39
Normal Probability Plot
Left-Skewed
Right-Skewed
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
-2 -1 0 1 2
Rectangular
U-Shaped
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
© 2003 Prentice-Hall, Inc.
Z
30
Z
30
-2 -1 0 1 2
Chap 5-40
Chapter Summary


Addressed the Probability of a Discrete
Random Variable
Defined Covariance and Discussed its
Application in Finance

Discussed Binomial Distribution

Discussed the Normal Distribution

Described the Standard Normal Distribution

Evaluated the Normality Assumption
© 2003 Prentice-Hall, Inc.
Chap 5-41