Probability Distributions: Binomial, Poisson, and Hypergeometric, Study notes of Mathematics

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Some Important Discrete

Probability Distributions

Topics

• The probability of a discrete random

variable

• Covariance and its applications in finance • Binomial distribution • Poisson distribution • Hypergeometric distribution

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 five times;

Count the number of tails(0, 1, 2, 3, 4, or 5 times)

Probability Distribution Values

Probability

0

1/4 =.

1

2/4 =.

2

1/4 =.

Discrete Probability

Distribution Example

Event: Toss 2 Coins.

Count # Tails.

T

T T

T

Summary Measures

Expected value (the mean)^ Weighted average of the probability

distribution:

(^

)^

j^

j

j

E

X

X P

X

Summary Measures

Example of expected value (the mean):

Toss two coins, count the number of

tails,

compute expected value

(^

)

(^

)(

)^

( )(

)^

(^

)(

)

0

1

.

2

.

1

j^

j

j

X P

X

=

=

∑

continued

Example of variance:

Toss two coins, count number of tails,compute variance

Summary Measures

(continued)

(^

)^

(^

)

(^

) (

)^

(^

) (

)^

(^

) (

)

2

2

2

2

2

0

1

.

1

1

.

2

1

.

.

j^

j

X

P

X

=

−

=

−

−

−

=

∑

Covariance and its Application

(^

)^

(^

)^

(^

)

(^

(^1) ) th th

th

: discrete random variable:

outcome of

: discrete random variable:

outcome of

: probability of occurrence of theoutcome of

an

N

XY

i^

i^

i^

i

i

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

=

=

−

−

⎡

⎤ ⎡

⎤

⎣

⎦ ⎣

⎦

∑

th

d the

outcome of Y

i

13

Computing the Variance for

Investment Returns

P(X

Yi^

)^ i

Economic condition

Dow Jones fund X

Growth Stock Y

.

Recession

-$

-$

.

Stable Economy

• 100

• 50

.

Expanding Economy

• 250

• 350

Investment

(^

)^

(^

)^

(^

2

2

2

.

100

105

.

100

105

.

250

105

14, 725

X

X

σ

σ

=

−

−

−

−

=

=

(^

)^

(^

)^

(^

2

2

2

2

.

200

90

.

50

90

.

350

90

37,

Y

Y

σ

σ

=

−

−

−

−

=

=

14

Computing the Covariance

for Investment Returns

P(X

Yi^

)^ i

Economic condition

Dow Jones fund X

Growth Stock Y

.

Recession

-$

-$

.

Stable Economy

• 100

• 50

.

Expanding Economy

• 250

• 350

Investment

(^

(^

)^

(^

(^

(^

(^

100

105

200

90

.

100

105

50

90

.

250

105

350

90

.

23,

XY σ

=

−

−

−

−

−

−

−

−

=

The Covariance of 23,000 indicates that the two investments arepositively related and will vary together in the same direction.

Binomial Probability Distribution • “n” Identical trials

• e.g.: 15 tosses of a coin; 10 light bulbs taken

from a warehouse

• Two mutually exclusive outcomes on each

trial^ – e.g.: Heads or tails 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 a coin is tossed

• Two sampling methods
  • Infinite population without replacement – Finite population with replacement

(continued)

19

Binomial

Distribution Characteristics

• Expected value (Mean)
  • – e.g.:
    • Variance and

standard deviation– – e.g.:

(^

)

E

X

np

μ =

=

(^

)

.

np

μ

=

=

=

n

= 5

p

= 0.

.6 .4 .2^0

0

1

2

3

4

5

X

P(X)

(^

)^

(^

)(

)

1

1

.

.

np

p

σ =

−

=

−

=

(^

)

(^

)

2

1 1

np

p

np

p

σ σ

=

−

=

−

Poisson Distribution

• Poisson process
  • Discrete events in an “interval”
    • The probability of one success

in an interval is stable

• The probability of more than

one success in this interval is 0

• The probability of success is

independent from interval to interval

• e.g.: The number of customers arriving in 15 minutes – e.g.: The number of defects per case of light bulbs

PX

x

x

x (^

|

!

=

λ

λ λ e