# Chap05 Discrete Probability Distribution

**Topics:**Random variable, Discrete probability distribution, Probability theory

**Pages:**50 (2676 words)

**Published:**March 29, 2015

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 )

i1

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 )

i1

Standard Deviation of a discrete random variable

σ σ2

N

2

[X

E(X)]

P(Xi )

i

i1

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 )

i1

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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