Module 9

1

• Topics
• Binomial Distribution
• Poisson Distribution
• Exponential Distribution
• Normal Distribution
• Beta Distribution

• Module 9
• Modeling Uncertainty
• THEORETICAL PROBABILITY MODELS

2
Introduction

• Module 7
• basic probability
• use in decision problems
• Module 8
• subjective probability
• modeling for decision analysis
• Module 9
• theoretical distributions
• application to decision analysis
• Module 9 software tutorial

3
Theoretical Probability ModelsLearning Objectives

• Refresh knowledge
• Binomial distribution
• Poisson distribution
• Exponential distribution
• Normal distribution
• Gain knowledge
• Beta distribution

4
Binomial Distribution(a discrete distribution)

• Model characteristics
• Dichotomous outcomes
• Two possible outcomes
• One outcome can occur
• Constant probability of success
• Independence

5
Binomial Distribution

• Mathematical model
• PB(R r n, p) n! / r! (n - r)! pr(1-p)n-r
• B subscript binomial probability
• R binomial random variable
• r number of successful outcomes
• n number of events or trials
• p probability of successful outcome
• E (R) np
• Var (R) np(1- p)

• 2

6
Binomial Distribution

• Probability mass function notation
• f (x) ( ) px qn-x
• X binomial random variable
• x number of successful outcomes
• n number of events or trials
• p probability of a successful outcome
• q 1 p probability of an unsuccessful
  outcome
• E (X) µ np
• V (X) s2 µq

n x
7
Binomial Distribution

• Cumulative distribution function
• Probability of k or fewer successful outcomes
• F (x k) ? ( ) px qn-x

k
n x
X 0
8
Binomial Distribution

• Frequent application situations
• Quality control
• Reliability
• Survey sampling
• Approximation by other models
• Poisson p ? 0 and n ? 8
• Normal p 0.5 and np gt 5, OR
• p gt 0.5 and nq gt 5

9
Binomial Distribution

• Example application
• Items are manufactured in large lots, from each
  of which
• twenty units are selected at random. The lot is
  accepted if the
• sample contains three or fewer defectives. If the
  production
• process yields, on the average, ten percent
  defectives, what is
• the probability of lot acceptance?

10
Binomial Distribution

• Formulation
• Determine the probability of three or fewer
• successes in 20 independent trials, each
• having 0.1 probability of success.
• F (x 3) ? ( ) (0.1)x (0.9)20-x 0.867

3
20 x
X 0
11
Poisson Distribution(a discrete distribution)

• Model characteristics
• Time interval or spatial region
• Probability of an event is small
• Events are independent
• Events take place at a constant rate

12
Poisson Distribution

• Mathematical model
• PP (X k m) (e m m k) / k !
• P subscript Poisson probability
• X Poisson random variable
• k number of events that occur
• e natural logs base, 2.718
• m mean number of event occurrences
• E (X) µ m
• Var (X) s2 m

13
Poisson Distribution

• Probability mass function notation
• f (x) (e ? ? x ) / x !
• x number of event occurrences
• ? mean number of event occurrences
• e natural logs base, 2.718
• E (X) µ ?
• V (X) s2 ?

14
Poisson Distribution

• Cumulative distribution function
• Probability of k or fewer occurrences within a
  temporal or spatial interval
• F (x k) ( e ? ? x ) / x !

? X 0
k
15
Poisson Distribution

• Frequent application situations
• Quality control
• Reliability
• Queuing
• Physical properties
• Approximation by another model
• Normal ? gt 5

16
Poisson Distribution

• Example application
• The probability that a person will have a
  negative
• reaction to the injection of a certain serum is
  0.001.
• Determine the probability that two or more of
  1,000
• people will have a negative reaction to the
  injection.

17
Poisson Distribution

• Formulation
• Determine the probability that zero or one
• people will have a negative reaction
• when ? np 1, and take the complement.
• 1 F (x 1) 1 - e-11x / x! 0.264

1
? x 0
18
Exponential Distribution(a continuous
distribution)

• Model characteristics
• Time or distance between two outcome occurrences
• Poisson model characteristics required
• Uses Poisson mean ( m or ? )

19
Exponential Distribution

• Mathematical model
• f E ( T t m ) me-mt
• E subscript exponential probability
• T exponential random variable
• t interarrival period
• m mean of Poisson distribution
• e natural logs base, 2.718..
• E (T) µ 1 / m
• Var (T) s2 1 / m2

20
Exponential Distribution

• Probability density function notation
• f (x) ? e -?x
• x length of interval between
• occurrences
• ? mean of Poisson distribution
• e natural logs base, 2.718
• E (X) µ 1 / ?
• V (X) s2 1 / ?2

21
Exponential Distribution

• Cumulative distribution function
• Probability that the interval between two
  occurrences is of length k or less
• F ( x k ) ? ? e ? x d x 1 e ? k

k
0
22
Exponential Distribution

• Frequent application situations
• Quality control
• Reliability
• Queuing
• Physical phenomena
• Applicable when underlying process is Poisson

23
Exponential Distribution

• Example Application
• The life of a certain brand of light bulb can be
  approximated
• by the exponential probability density function.
  If the mean
• life of the light bulb is 1,000 hours, determine
  the probability
• that the bulb will last more than 1,000 hours

24
Exponential Distribution

• Formulation
• Determine the probability that a light bulb will
  have a life of 1000 or fewer hours, when µ
  1000, and take the complement
• µ 1/? 1000 so ? 1/1000 0.001
• 1 F(x 1000) 1 - ? 0.001 e- 0.001x dx
• 1 - 1- e-0.001(1000) 1 - 1 e-1
  0.368

1000
0
25
Normal (Gaussian) Distribution(a continuous
distribution)

• Model characteristics
• Bellshaped curve
• Effective for measured phenomena
• Effective for multiple sources of uncertainty
• Two parameters, µ and s

26
Normal Distribution

• Mathematical model
• fn ( y µ,s2 ) (2ps2)-1/2 exp -(y-µ)2 / 2s2
• N subscript normal probability
• y value(s) of random variable Y
• p 3.14159
• e natural logs base 2.718
• µ the distribution mean
• s the distribution standard deviation
• E (Y) µ
• Var (Y) s2

27
Normal Distribution

• Probability density function
• f (x) (2ps2)-1/2 exp -(x-µ)2 / 2s2
• - 8 lt x lt 8
• - 8 lt µ lt 8
• - s gt 0
• - p, e are constants
• - µ, s are parameters
• E (X) µ
• Var (X) s2

28
Normal Distribution

• Cumulative distribution function
• Probability that a value of x or less will occur
• F (x) (2ps2)-1/2 ? exp -(v-µ)2 / 2s2 dv
• P (a lt X b) F (b) F (a)
• (2ps2)-1/2? exp -(v-µ)2 / 2s2 dv

x
-8

b
a
29
Normal Distribution

• Frequent application situations
• measured phenomena
• measured results of multiple additive
  phenomena
• approximation of other distributions
• numerous inferential statistical methods

30
Normal Distribution

• Example application
• The ages of employees at a company are
• normally distributed with a mean of 50 years
• and a standard deviation of 5 years. Determine
• the percentage of employees whose ages are
• likely to be between 50 and 52.5 years.

31
Normal Distribution

• Formulation
• Determine the probability of age 50 years, and
  determine the probability of age 52.5 years.
  Take the difference, and multiply the
• result by 100 to obtain the percentage.
• Using the standard normal variate, Z (x µ) /
  s, and cumulative distribution function
  probabilities from either tabled values or
  appropriate software, we obtain
• F(x 50) F(Z 0) 0.5000 and
• F(x 52.5) F(Z 0.5) 0.6915
• Then
• P(50 lt x 52.5) 0.6915 0.5000 0.1915 or
  19.15

32
Beta Distribution

• Model characteristics
• Variable bounded at both ends
• Numerous distribution shapes
• Events are independent
• Models proportions

33
Beta Distribution

• Generalized beta probability density function
• f (x ?1, ?2, ?1, ?2)
• where
• - ?1 x ?2
• - ?1, ?2, 0
• - G(?) (? 1)! when ? ? I, OR

(
)
(
)
?2-1
1
?1-1
(?1?2)
G
____
_________
______
x ?1
_____
x ?1
1 -
?2 ?1
?2 ?1
?2-?1
G
(?1)
G
(?2)
8
? x?-1 e-x dx otherwise
0
34
Beta Distribution

• More frequently,
• set ?1 0 and ?2 1
• So f (x ?1, ?2)
  x?1-1 (1-x)?2-1
• Where 0 x 1
• ?1, ?2 0
• G (?) (?-1)! Where ? ? I, OR

__________
G (?1 ?2)
G (?1) G (?2)
8
? x?-1 e-x dx otherwise
0
35
Beta Distribution

• When ?1 and ?2 are positive integers
• Probability density function
• f (x ?1, ?2) x ?1-1 (1 x)?2-1
• where the random variable X x
• and 0 x 1

(?1 ?2 1)!
_____________
(?1 1)! (?2 1)!
36
Beta Distribution

• Let n ?1 ?2 and r ?1
• f ß ( q r,n )
  q r-1 ( 1 q )n-r-1
• ß subscript beta probability
• Q Beta random variable
• q proportion value between 0 and 1
• r number of successes
• n number of trials
• E (Q) µ r / n
• Var (Q) s2 r ( n r) / n2 ( n 1)

________________
(n 1)!
( r 1) ! ( n r 1) !
37
Beta Distribution

• Cumulative distribution function
• F (Q q r,n) ? ?r-1 (1 ?) d?
• Called Incomplete Beta Function
• Equivalence to binomial cumulative distribution
  function
• Values from tables or software

______________
(n 1)!
q
(r 1)! (n r 1)
0
38
Beta Distribution

• Frequent application situations
• Proportions or percentages
• Physical variables in restricted intervals
• Tolerances in quality control and reliability
• PERT networks (generalized form)
• Bayesian analysis (informative a priori)
• Approximated by difference between two normal
  distributions when ?1 ?2 30

39
Beta Distribution

• Example Application
• Suppose that the percentage of employees who
  submit medical benefits claims each year is a
  beta random variable. As the benefits coordinator
  for a small firm with 40 employees, you currently
  expect that about 30 percent will most likely
  submit claims this year. Determine the
  probability that more than 16 employees will
  submit claims.

40
Beta Distribution

• Formulation
• Determine the probability that 16 or fewer of 40
  employees will submit claims, and take the
  complement.
• µ r/n 0.30 r (0.30)(40) 12
• q 16/40 0.40
• From table values or software,
• Fß (Q 0.40 12, 40) 0.91
• Thus Pß (Q gt 0.40) 1 0.91 0.09

41
Summary

• Theoretical distributions to model uncertainty
• Five distributions
• Binomial
• Poisson
• Exponential
• Normal
• Beta
• Each distribution
• Mathematical formulation
• Mean and variance
• Graphical representations
• Application situations
• Example problem