Chap05 Discrete Probability Distribution

Topics: Random variable, Discrete probability distribution, Probability theory Pages: 50 (2676 words) Published: March 29, 2015
Business Statistics
Chapter 5
Some Important Discrete
Probability Distributions

5-1

Chapter Goals
After completing this chapter, you should be able
to:
 Interpret the mean and standard deviation for a
discrete probability distribution
 Explain covariance and its application in finance
 Use the binomial probability distribution to find
probabilities
 Describe when to apply the binomial distribution
 Use Poisson discrete probability distributions to
find probabilities
5-2

Definitions
Random Variables
 A random variable represents a possible
numerical value from an uncertain event.
 Discrete random variables produce outcomes
that come from a counting process (e.g. number
of courses you are taking this semester).
 Continuous random variables produce outcomes
that come from a measurement (e.g. your annual
salary, or your weight).
5-3

Definitions
Random Variables
Random
Variables
Ch. 5

Discrete
Random Variable

Continuous
Random Variable

Ch. 6

5-4

Discrete Random Variables
 Can only assume a countable number of values
Examples:
 Roll a die twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)

 Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
5-5

Probability Distribution for a
Discrete Random Variable
 A probability distribution (or probability mass function )(pdf) for a discrete random variable is a mutually exclusive listing of all possible numerical outcomes for that random variable
such that a particular probability of occurrence is associated with each outcome.
Number of Classes
Taken

Probability

2

0.2

3

0.4

4

0.24

5

0.16
5-6

Discrete Probability Distribution
Experiment: Toss 2 Coins.

T
T
H
H

T
H
T
H

Probability Distribution
X Value

Probability

0

1/4 = .25

1

2/4 = .50

2

1/4 = .25

Probability

4 possible outcomes

Let X = # heads.

.50
.25

0

1

2

X
5-7

Discrete Random Variable
Summary Measures
 Expected Value (or mean) of a discrete
distribution (Weighted Average)
N

 E(X)  Xi P( Xi )
i1

 Example: Toss 2 coins,
X = # of heads,
compute expected value of X:

X

P(X)

0

.25

1

.50

2

.25

E(X) = (0 x .25) + (1 x .50) + (2 x .25)
= 1.0
5-8

Discrete Random Variable
Summary Measures
(continued)

 Variance of a discrete random variable
N

σ 2  [Xi  E(X)]2 P(Xi )
i1

 Standard Deviation of a discrete random variable

σ  σ2 

N

2
[X

E(X)]
P(Xi )
 i
i1

where:
E(X) = Expected value of the discrete random variable X
Xi = the ith outcome of X
P(Xi) = Probability of the ith occurrence of X
5-9

Discrete Random Variable
Summary Measures
(continued)

 Example: Toss 2 coins, X = # heads,
compute standard deviation (recall E(X) = 1)

σ

2

 [X  E(X)] P(X )
i

i

σ  (0  1)2 (.25)  (1  1)2 (.50)  (2  1)2 (.25)  .50 .707 Possible number of heads
= 0, 1, or 2

5-10

The Covariance
 The covariance measures the strength of the
linear relationship between two variables
 The covariance:
N

σ XY  [ Xi  E( X)][( Yi  E( Y )] P( Xi Yi )
i1

where:

X = discrete variable X
Xi = the ith outcome of X
Y = discrete variable Y
Yi = the ith outcome of Y
P(XiYi) = probability of occurrence of the condition affecting the ith outcome of X and the ith outcome of Y
5-11

Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments

P(XiYi)

Economic condition

Investment
Passive Fund X Aggressive Fund Y

.2

Recession

- $ 25

- $200

.5

Stable Economy

+ 50

+ 60

.3

Expanding Economy

+ 100

+ 350

E(X) = μX = (-25)(.2) +(50)(.5) + (100)(.3) = 50
E(Y) = μY = (-200)(.2) +(60)(.5) + (350)(.3) = 95
5-12

Computing the Standard Deviation
for Investment Returns
P(XiYi)

Economic condition

Investment...
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