© 2003 Prentice-Hall, Inc. Chap 5-1
Business Statistics: A First Course
(3rd Edition)
Chapter 5Probability Distributions
© 2003 Prentice-Hall, Inc. Chap 5-2
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 5-3
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 5-4
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 5-5
Probability Distribution Values Probability
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
Discrete Probability Distribution Example
Event: Toss 2 Coins. Count # Tails.
T
T
T T
© 2003 Prentice-Hall, Inc. Chap 5-6
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 1 1j jP X P X
© 2003 Prentice-Hall, Inc. Chap 5-7
Summary Measures
Expected value (The Mean) Weighted average of the probability
distribution
e.g. Toss 2 coins, count the number of tails, compute expected value
j jj
E X X P X
0 .25 1 .5 2 .25 1
j jj
X P X
© 2003 Prentice-Hall, Inc. Chap 5-8
Summary Measures
Variance Weighted average squared deviation about the
mean
e.g. Toss 2 coins, count number of tails, compute variance
(continued)
222j jE X X P X
22
2 2 2 0 1 .25 1 1 .5 2 1 .25 .5
j jX P X
© 2003 Prentice-Hall, Inc. Chap 5-9
Covariance and its Application
1
th
th
th
: discrete random variable
: outcome of
: discrete random variable
: outcome of
: probability of occurrence of the
outcome of an
N
XY i i i ii
i
i
i i
X E X Y E Y P X Y
X
X i X
Y
Y i Y
P X Y i
X
thd the outcome of Yi
© 2003 Prentice-Hall, Inc. Chap 5-10
Computing the Mean for Investment Returns
Return per $1,000 for two types of investments
P(Xi) P(Yi) 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
Investment
100 .2 100 .5 250 .3 $105XE X
200 .2 50 .5 350 .3 $90YE Y
© 2003 Prentice-Hall, Inc. Chap 5-11
Computing the Variance for Investment Returns
2 2 22 .2 100 105 .5 100 105 .3 250 105
14,725 121.35X
X
2 2 22 .2 200 90 .5 50 90 .3 350 90
37,900 194.68Y
Y
P(Xi) P(Yi) 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
Investment
© 2003 Prentice-Hall, Inc. Chap 5-12
Computing the Covariance for Investment Returns
P(XiYi) Economic condition Dow Jones fund X Growth Stock Y
.2 Recession -$100 -$200
.5 Stable Economy + 100 + 50
.3 Expanding Economy + 250 + 350
Investment
100 105 200 90 .2 100 105 50 90 .5
250 105 350 90 .3 23,300
XY
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 5-13
Important Discrete Probability Distributions
Discrete Probability Distributions
Binomial
© 2003 Prentice-Hall, Inc. Chap 5-14
Binomial Probability Distribution
‘n’ Identical Trials e.g. 15 tosses of a coin; 10 light bulbs taken
from a warehouse 2 Mutually Exclusive Outcomes on Each
Trial 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 5-15
Binomial Probability Distribution
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
(continued)
© 2003 Prentice-Hall, Inc. Chap 5-16
Binomial Probability Distribution Function
!1
! !
: probability of successes given and
: number of "successes" in sample 0,1, ,
: the probability of each "success"
: sample size
n XXnP X p p
X n X
P X X n p
X X n
p
n
Tails in 2 Tosses of Coin
X P(X) 0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
© 2003 Prentice-Hall, Inc. Chap 5-17
Binomial Distribution Characteristics
Mean E.g.
Variance and Standard Deviation
e.g.
E X np 5 .1 .5np
n = 5 p = 0.1
0.2.4.6
0 1 2 3 4 5
X
P(X)
1 5 .1 1 .1 .6708np p
2 1
1
np p
np p
© 2003 Prentice-Hall, Inc. Chap 5-18
Binomial Distribution in PHStat
PHStat | Probability & Prob. Distributions | Binomial
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 5-19
Continuous Probability Distributions
Continuous Random Variable Values from interval of numbers Absence of gaps
Continuous Probability Distribution Distribution of continuous random variable
Most Important Continuous Probability Distribution The normal distribution
© 2003 Prentice-Hall, Inc. Chap 5-20
The Normal Distribution
“Bell Shaped” Symmetrical Mean, Median and
Mode are Equal Interquartile Range
Equals 1.33 Random Variable
has Infinite Range
Mean Median Mode
X
f(X)
© 2003 Prentice-Hall, Inc. Chap 5-21
The Mathematical Model
2(1/ 2) /1
2
: density of random variable
3.14159; 2.71828
: population mean
: population standard deviation
: value of random variable
Xf X e
f X X
e
X X
© 2003 Prentice-Hall, Inc. Chap 5-22
Many Normal Distributions
Varying the Parameters and , we obtain Different Normal Distributions
There are an Infinite Number of Normal Distributions
© 2003 Prentice-Hall, Inc. Chap 5-23
Finding Probabilities
Probability is the area under the curve!
c dX
f(X)
?P c X d
© 2003 Prentice-Hall, Inc. Chap 5-24
Which Table to Use?
Infinitely Many Normal Distributions Mean Infinitely Many Tables to Look
Up!
© 2003 Prentice-Hall, Inc. Chap 5-25
Solution: The Cumulative Standardized Normal
Distribution
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5478.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
Probabilities
Only One Table is Needed
0 1Z Z
Z = 0.12
0
© 2003 Prentice-Hall, Inc. Chap 5-26
Standardizing Example
6.2 50.12
10
XZ
Normal Distribution
Standardized Normal
Distribution10 1Z
5 6.2 X Z
0Z 0.12
© 2003 Prentice-Hall, Inc. Chap 5-27
Example:
Normal Distribution
Standardized Normal
Distribution10 1Z
5 7.1 X Z0Z
0.21
2.9 5 7.1 5.21 .21
10 10
X XZ Z
2.9 0.21
.0832
2.9 7.1 .1664P X
.0832
© 2003 Prentice-Hall, Inc. Chap 5-28
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5832.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = 0.21
Example: 2.9 7.1 .1664P X
(continued)
0
© 2003 Prentice-Hall, Inc. Chap 5-29
Z .00 .01
-0.3 .3821 .3783 .3745
.4207 .4168
-0.1.4602 .4562 .4522
0.0 .5000 .4960 .4920
.4168.02
-0.2 .4129
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = -0.21
Example: 2.9 7.1 .1664P X
(continued)
0
© 2003 Prentice-Hall, Inc. Chap 5-30
Normal Distribution in PHStat
PHStat | Probability & Prob. Distributions | Normal …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 5-31
Example: 8 .3821P X
Normal Distribution
Standardized Normal
Distribution10 1Z
5 8 X Z0Z
0.30
8 5.30
10
XZ
.3821
© 2003 Prentice-Hall, Inc. Chap 5-32
Example: 8 .3821P X
(continued)
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.6179.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = 0.30
0
© 2003 Prentice-Hall, Inc. Chap 5-33
.6217
Finding Z Values for Known Probabilities
Z .00 0.2
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
.6179 .6255
.01
0.3
Cumulative Standardized Normal Distribution Table
(Portion)
What is Z Given Probability = 0.6217 ?
.6217
0 1Z Z
.31Z 0
© 2003 Prentice-Hall, Inc. Chap 5-34
Recovering X Values for Known Probabilities
5 .30 10 8X Z
Normal Distribution
Standardized Normal
Distribution10 1Z
5 ? X Z0Z 0.30
.3821.6179
© 2003 Prentice-Hall, Inc. Chap 5-35
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 5-36
Assessing Normality Construct Charts
For small- or moderate-sized data sets, do stem-and-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 ?
(continued)
© 2003 Prentice-Hall, Inc. Chap 5-37
Assessing Normality
Observe the Distribution of the Data Set Do approximately 2/3 of the observations lie
between mean 1 standard deviation? Do approximately 4/5 of the observations lie
between mean 1.28 standard deviations? Do approximately 19/20 of the observations
lie between mean 2 standard deviations? Evaluate Normal Probability Plot
Do the points lie on or close to a straight line with positive slope?
(continued)
© 2003 Prentice-Hall, Inc. Chap 5-38
Assessing Normality
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
(continued)
© 2003 Prentice-Hall, Inc. Chap 5-39
Assessing Normality
Normal Probability Plot for Normal Distribution
Look for a Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
(continued)
© 2003 Prentice-Hall, Inc. Chap 5-40
Normal Probability Plot
Left-Skewed Right-Skewed
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
© 2003 Prentice-Hall, Inc. Chap 5-41
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