Transcript Probability Distributions

BA 201
Lecture 7
The Probability Distribution for a
Discrete Random Variable
© 2001 Prentice-Hall, Inc.
Chap 5-1
Topics

The Probability of a Discrete Random Variable
© 2001 Prentice-Hall, Inc.
Chap 5-2
Population and Sample
Population
p.??
Sample
Use statistics to
summarize features
Use parameters to
summarize features
Inference on the population from the sample
© 2001 Prentice-Hall, Inc.
Chap 5-3
p.9
Types of Data
Data
Categorical
(Qualitative)
Numerical
(Quantitative)
Discrete
(counting)
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Continuous
(measurement)
Chap 5-4
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)
E.g. Toss a coin; assign $10 to head and -$30 to a
tail
= $10
© 2001 Prentice-Hall, Inc.
T
= -$30
Chap 5-5
Discrete Random Variable

Discrete Random Variable

Obtained by Counting (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)
© 2001 Prentice-Hall, Inc.
Chap 5-6
Discrete Probability Distribution
Example
Event: Toss 2 Coins.
Count # Tails.
Probability Distribution
Values
Probability
T
T
T
© 2001 Prentice-Hall, Inc.
T
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
This is using the A Priori Classical
Probability approach.
Chap 5-7
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
© 2001 Prentice-Hall, Inc.
PX  1
j
Chap 5-8
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
© 2001 Prentice-Hall, Inc.
  0 .25  1.5   2 .25  1
Chap 5-9
Summary Measures

(continued)
Variance



Weight 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
© 2001 Prentice-Hall, Inc.
2
2
Chap 5-10
Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments
Investment
Economic condition Dow Jones fund X Growth Stock Y
P(XiYi)
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.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
© 2001 Prentice-Hall, Inc.
Chap 5-11
Computing the Variance for
Investment Returns
Investment
Economic condition Dow Jones fund X Growth Stock Y
P(XiYi)
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.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
© 2001 Prentice-Hall, Inc.
2
2
2
 Y  194.68
Chap 5-12
p.183
Computing the Coefficient of
Variation for Investment Returns




 X 121.35
CV  X  

 1.16  116%
X
105
 Y 194.68
CV Y  

 2.16  216%
Y
90
Investment X appears to have a lower risk
(variation) per unit of average payoff (return)
than investment Y
Investment X appears to have a higher
average payoff (return) per unit of variation
(risk) than investment Y
© 2001 Prentice-Hall, Inc.
Chap 5-13
pp.246-247
Doing It in PHStat

PHStat | Probability Distributions |
Covariance and Portfolio Analysis



Fill in the probabilities and outcomes for
investment X and Y
Manually compute the CV using the formula
Here is the Excel spreadsheet that contains
the results of the previous investment
example.

© 2001 Prentice-Hall, Inc.
Chap 5-14
Summary

Addressed the Probability of a Discrete
Random Variable
© 2001 Prentice-Hall, Inc.
Chap 5-15