MBA3 Probability distributions and information

1
MBA3 Probability distributions and
information

• Fred Wenstøp

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Discrete probability distributions

• A series of probabilities pi for all possible
  states of nature
• Spi 1
• A probability distribution can be
• Theoretical, based on simple but fundamental
  assumptions
• Binomial
• Pascal
• Poisson
• Empirical, based on past experience
• Subjective, based on beliefs

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The binomial distribution

• How many times will I succeed?
• A binomial process
• A series of n independent trials where the
  outcome each time is either success or failure
  and a constant probability p for success
• The probability of exactly a successes in a
  binomial process
• Excel
• BINOMDIST(anp0)

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The Pascal distribution

• When will it be my turn?
• The probability that it will take n trials to get
  the first success in a binomial process with
  probability p.
• Example
• How many tosses to get on the board i Ludo?
• p 1/6
• See graph

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The Poisson distribution

• How often will disasters happen?
• The probability of x occurrences of an event in a
  certain period when the propensity for the event
  to occur is constant and equal to l per period.
• POISSON(xl0)
• Example
• Norway has about two oil spills per year in
  coastal waters. The probability that we will have
  x spills in a certain year is
• POISSON(x20)

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Empirical or subjective distributions

• Based on experience or merely assumed
• Example
• Future sales of a consumer good

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Continuous probability distributionsProbability
densities

• In many situations, any value among an infinite
  number can in principle occur
• In practice, the number depends on how precisely
  we measure the differences between them
• Future stock price
• Future employment rates
• Future sales
• The probability of a particular value is
  therefore zero
• Instead, we use probability densities, where
  areas are probabilities
• Example The normal distribution

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Cumulative distributions

• Probability densities can be represented as
  cumulative distributions which make them easier
  to handle
• The probability of at least x
• y F(x)
• F(a) P(xlta)
• Important parameters
• Median m, F(m) 0.5
• Fractiles
• The median is the 50 fractile

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Conditional probabilityThe value of tests

• A production process produces defect units with
  probability 0.1
• If an OK unit is shipped, the reward is 100
• If it is defect, a loss of 160 is incurred
• A unit can be reworked at an expense of 40 and
  becomes OK regardless of previous state
• A test with sensitivity 0.7 and specificity 0.8
  may be performed before any decision is made
• What should you be willing to pay for the test?

Production P(OK) 0.9 P(D) 0.1 Test P(TDD)
0.7 P(TOKD) 0.3 P(TOKOK) 0.8 P(TDOK) 0.2
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Probability tree to represent conditional
probabilities
0.72
Production P(OK) 0.9 P(D) 0.1 Test P(TDD)
0.7 P(TOKD) 0.3 P(TOKOK) 0.8 P(TDOK) 0.2
TOK
0.8
TD
OK
0.18
0.2
0.9
0.03
D
TOK
0.1
0.3
TD
0.7
0.07
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Transforming a probability tree to a decision
oriented tree
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Decision analysis
Rework
60
60
D 0.1
-160
Ship
D 0.04
OK 0.9
-160
82.2
100
74
OK 0.96
89.6
100
Ship
Rework
60
82.2
TOK 0.75
89.6
D 0.28
-160
Test
OK 0.72
Ship
27.2
100
TD 0.25
Rework
60
60