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ECO 321 Intermediate Business Statistics

Lesson 5

Expectation, Their Applications and Some Important Probability

Distributions

Lesson Topics

 The Probability Distribution of a Discrete

Random Variable

 Covariance and its Applications in Finance

 The Binomial Distribution

 The Normal Distribution

 The Standardized Normal Distribution

 Evaluating the Normality Assumption

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); measuring

the amount of rainfall on a particular day

 E.g. Toss a coin; assign $10 to head and -$30 to a

tail = $10

T = -$30

Random Variables and Data

Random Variables

Discrete Random Variables Continuous Random Variables

Discrete Data Continuous Data

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)

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 This is using the A Priori Classical Probability approach.

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  

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 j j

E X X P X   

 

            0 . 2 5 1 . 5 2 . 2 5 1

j j

X P X 

   



Summary Measures

 Variance

 Weighted average squared deviation about the mean



 E.g. Toss 2 coins, count number of tails, compute variance

(continued)

      222

j j E X X P X        



   

           

2 2

2 2 2

0 1 . 2 5 1 1 . 5 2 1 . 2 5

. 5

j j X P X  

     





Covariance and its Application

     

 

t h

: d i s c r e t e r a n d o m v a r i a b le

: o u t c o m e o f

: d i s c r e t e r a n d o m v a r i a b le

: o u t c o m e o f

: p r o b a b i li t y o f o c c u r r e n c e o f t h e

o u t c o m e o f a n

X Y i i i i

i i

X E X Y E Y P X Y

X i X

Y i Y

P X Y i





         

t h d t h e o u t c o m e o f Yi

    1

A ls o

i X i Y

X Y

 







 

 





X 0

Y 

    0 X Y

X Y   

    0 X Y

X Y        0 X Y

X Y   

    0 X Y

X Y   

    1

A ls o

i X i Y

X Y

 







 

 





X 0

Y 

    0 X Y

X Y   

    0 X Y

X Y        0 X Y

X Y   

    0 X Y

X Y   

    1

A ls o

i X i Y

X Y

 







 

 





X 0

Y 

    0 X Y

X Y   

    0 X Y

X Y        0 X Y

X Y   

    0 X Y

X Y   

Computing the Mean for

Investment Returns

Return per $1,000 for two types of investments

             1 0 0 .2 1 0 0 .5 2 5 0 .3 $ 1 0 5 X

E X      

             2 0 0 .2 5 0 .5 3 5 0 .3 $ 9 0 Y

E 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

Computing the Variance for

Investment Returns

            2 2 22

. 2 1 0 0 1 0 5 . 5 1 0 0 1 0 5 . 3 2 5 0 1 0 5

1 4 , 7 2 5 1 2 1 . 3 5



      

 

            2 2 22

. 2 2 0 0 9 0 . 5 5 0 9 0 . 3 3 5 0 9 0

3 7 , 9 0 0 1 9 4 . 6 8



      

 

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

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

           

     

1 0 0 1 0 5 2 0 0 9 0 . 2 1 0 0 1 0 5 5 0 9 0 . 5

2 5 0 1 0 5 3 5 0 9 0 . 3 2 3 , 3 0 0

X Y         

   

The Covariance of 23,000 indicates that the two investments are

positively related and will vary together in the same direction.

Computing the Coefficient of Variation for Investment Returns



 Investment X appears to have a lower risk (variation) per unit of average payoff (return) than investment Y

  1 2 1 . 3 5

1 . 1 6 1 1 6 % 1 0 5

C V X 

    

  1 9 4 . 6 8

2 . 1 6 2 1 6 % 9 0

C V Y 

    

Investment Decisions

Risk Expected Return

Trade-off between risk (standard deviation) and expected return (average)

1. The ________________measures the absolute risk

2. The relative risk is measured by the __________________

standard deviation

coefficient of variation

Different Types of Investors

 If you are a “risk averse” investor (i.e. you don’t like risk but would like a high expected return), you will choose Stock A because it has the lowest risk per dollar of average return even though it does not have the highest expected return and the lowest absolute risk. (Uses CV)

 If you are a “risk seeking” investor (i.e. you love risk and would like a high expected return), you will choose Stock B because it has both the highest risk and average return. (Wants both high)

 If you are a “risk neutral” investor (i.e. you don’t care about risk but would like a high expected return), you will choose Stock B because it has the highest average return regardless of the risk. (Ignores risk)

Procedure for Various Types of Investors in Making Investment Decision

 Risk Averse Investors

 Pick the stock that gives the highest average return and the lowest absolute risk

 If such a stock does not exist, pick the one that gives the lowest relative risk

 Risk Seeking Investors

 Pick the stock that gives the highest average return and the highest absolute risk

 If such a stock does not exist, pick the one that gives the highest relative risk

 Risk Neutral Investors

 Pick the stock that gives the highest average return regardless of the absolute or relative risk

Sum of Two Random Variables

 The expected value of the sum is equal to the sum of the expected values

 The variance of the sum is equal to the sum of the variances plus twice the covariance

 The standard deviation is the square root of the variance

     E X Y E X E Y  

  2 2 2

2 X Y X Y X Y

V a r X Y     

    

X Y X Y  

  

Portfolio Expected Return and Risk

 The portfolio expected return for a two-asset investment is equal to the weighted average of the two assets

 Portfolio risk

       1

w h e r e

p o r t i o n o f t h e p o r t f o l i o v a l u e a s s i g n e d t o a s s e t

E P w E X w E Y

w X

  



    22 2 2

1 2 1 P X Y X Y

w w w w       

Computing the Expected Return and

Risk of the Portfolio Investment

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

Suppose a portfolio consists of an equal investment in each of X and Y.

     0 .5 1 0 5 0 .5 9 0 9 7 .5E P   

              2 2

0 . 5 1 4 7 2 5 0 . 5 3 7 9 0 0 2 0 . 5 0 . 5 2 3 3 0 0 1 5 7 . 5 P

    

Computing the Expected Return and

Risk of the Portfolio Investment Covariance Analysis

Probabilities & Outcomes: P X Y

0.2 -100 450

0.5 100 60

0.3 250 -100

Weight Assigned to X 0.5

Calculations Area

E(X) 105

E(Y) 90

Variance(X) 14725

Standard Deviation(X) 121.3466

Variance(Y) 37200

Standard Deviation(Y) 192.873

Covariance(XY) -22950

Variance(X+Y) 6025

Standard Deviation(X+Y) 77.62087

Weight Assigned to X 0.5

Weight Assigned to Y 0.5

Portfolio Expected Return 97.5

Statistics

Portfolio Management

Using PHStat

 PHStat | Decision Making | Covariance and Portfolio Analysis  Fill in the “Number of Outcomes:”

 Check the “Portfolio Management Analysis” box

 Fill in the probabilities and outcomes for investment X and Y

 Manually compute the CV using the formula in the previous slide

 Here is the Excel spreadsheet that contains the results of the previous investment example. 

 See PHStat Animation #9

Properties of 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 Trials

 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

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)

Binomial Probability Distribution Function

   

 

! 1

! !

: p r o b a b i li t y o f s u c c e s s e s g i v e n a n d

: n u m b e r o f " s u c c e s s e s " i n s a m p le 0 , 1, ,

: t h e p r o b a b i li t y o f e a c h " s u c c e s s "

: s a m p le s i z e

n XX n

P X X n X

P X X n

X X n

 





  



E.g. Tails in 2 Tosses of Coin

X P(X)

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Binomial Distribution Characteristics

 Mean



 E.g.

 Variance and Standard Deviation



 E.g.

 E X n  

 5 , .1, 5 .1 .5n n      

n = 5 p = 0.1

0 1 2 3 4 5

P(X)

     1 5 .1 1 .1 .6 7 0 8n      

 

2 1

  

 

5 , .1n  

Binomial Distribution in PHStat

 PHStat | Probability & Prob. Distributions | Binomial

 Example in Excel Spreadsheet when n=5, p=.1

More Binomial Examples

A mid-term exam has 30 multiple-choice questions, each with 5 possible answers. What is the probability of randomly guessing the answer for each question and passing the exam (i.e., having guessed at least 18 questions correctly).

Are the assumptions for the binomial distribution met?

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

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

f(X)



The Mathematical Model (Called the Probability Density Function)

   

 

: d e n s i t y o f r a n d o m v a r i a b le

3 . 1 4 1 5 9 ; 2 . 7 1 8 2 8

: p o p u la t i o n m e a n

: p o p u la t i o n s t a n d a r d d e v i a t i o n

: v a lu e o f r a n d o m v a r i a b le

f X e

f X X

X X

 

 







 



 

    

Many Normal Distributions

Varying the Parameters X and X, We Obtain Different Normal Distributions

There are an Infinite Number of Normal Distributions

The Standardized Normal Distribution

 Given , has a

standardized (normalized) normal distribution

where

  2

, X X

X N   X

X Z





 

f(X)

X 

X 

0 Z

 

1 Z

 

f(Z)

 0 , 1Z N

Finding Probabilities

Probability is the area under the curve!

c d X

f(X)

  ?P c X d  

a b

Example:

Normal Distribution Standardized

Normal Distribution

Shaded Area Exaggerated

1 0 X

  1

Z  

5 X

 

7 .1 X Z 0

Z  

0 .2 1

2 . 9 5 7 . 1 5 . 2 1 . 2 1

1 0 1 0

X X

X X Z Z

 

 

          

2 .9 0 .2 1

.0 8 3 2

 2 .9 7 .1 .1 6 6 4P X  

.0 8 3 2

Example: or

 What is the value of a ?

  .6 2 1 7P X a 

.6217

Shaded Area Exaggerated

0 1 Z Z

  

.3 1Z 

.6217

5 1 0 X X

  

8 .1a 

Normal Distribution Standardized

Normal Distribution

X Z

  .3 7 8 3P X a 

Normal Distribution in PHStat

 PHStat | Probability & Prob. Distributions | Normal …

 Example in Excel Spreadsheet

 See PHStat Animation #12

More Example on Normal Distribution

A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8.

What is the probability of getting a grade no higher than 91 on this exam?

  2

7 3, 8X N  9 1 ?P X  

Mean 73

Standard Deviation 8

X Value 91

Z Value 2.25

P(X<=91) 0.9877756

Probability for X <=

2.250

Z 73 91

What percentage of students scored between 65 and 89?

From X Value 65

To X Value 89

Z Value for 65 -1

Z Value for 89 2

P(X<=65) 0.1587

P(X<=89) 0.9772

P(65<=X<=89) 0.8186

Probability for a Range

  2

7 3, 8X N  6 5 8 9 ?P X  

Z 8965

-1

Only 5% of the students taking the test scored higher than what grade?

  2

7 3, 8X N  ? .0 5P X 

Cumulative Percentage 95.00%

Z Value 1.644853

X Value 86.15882

Find X and Z Given Cum. Pctage.

1.6450

Z 73 ? =86.16

The middle 50% of the students scored between what two scores?

  2

7 3, 8X N

Cumulative Percentage 75.00%

Z Value 0.67449

X Value 78.39592

Find X and Z Given Cum. Pctage.

Cumulative Percentage 25.00%

Z Value -0.67449

X Value 67.60408

Find X and Z Given Cum. Pctage.

0.670

Z 78.467.6

-0.67

  .5 0P a X b  

.25.25

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

Assessing Normality

 Construct Charts

 For small- or moderate-sized data sets, does box- and-whisker plot look symmetric?

 Does the histogram look symmetric?

 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)

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)



Assessing Normality

Normal Probability Plot for Normal Distribution

Look for Straight Line!

-2 -1 0 1 2

(continued)

Normal Probability Plot

Left-Skewed Right-Skewed

Rectangular U-Shaped

-2 -1 0 1 2

Obtaining Normal Probability Plot in PHStat

 PHStat | Probability & Prob. Distributions | Normal Probability Plot

 Enter the range of the cells that contain the data in the Variable Cell Range window

 See PHStat Animation #13

 Data on Body Fat

 Data on NFL Scores

excel/bodyfat.xls

pro_football.xls

Lesson Summary

 Addressed the Probability Distribution of a

Discrete Random Variable

 Defined Covariance and Discussed its

Application in Finance

 Discussed the Binomial Distribution

 Discussed the Normal Distribution

 Described the Standard Normal Distribution

 Evaluated the Normality Assumption