Probability and Statistics Todays Goals

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Probability and StatisticsTodays Goals

• Why do we need to make decisions under
  uncertainty?
• Randomness
• Measurement errors
• Understand how probability and statistics can
  support better decision making
• Apply the basic axioms of probability
• Homework Due Fri. Feb. 6. (because of snow day)
  Problems 8, 12, 18 in Chapter 2.1,2.2. PLUS, 3
  additional problems on web.

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Uncertainty

• Randomness (stochasticity)
• Measurement Error (parameter uncertainty)

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Stochasticity
Bars dust storm index line rainfall
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Stochasticity
Yield strength histogram of steel grade Q235 .
Steel Plate Used for Penstocks of Hydropower
Stations
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Measurement Error

• We believe that there is a fixed relationship,
  but we cannot measure it accurately.
• Example Climate Sensitivity
• Example Resistance

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What do we do when faced with uncertainty?

• How do you design a policy for climate change?
• How do you design a culvert for flood prevention?

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What do we do when faced with uncertainty?

• How do you design a policy for climate change?
• How do you design a culvert for flood prevention?
• Plan for the worst case?
• Plan for the average case?

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Examples

• Planning and design of airport pavement
• Thicker lasts longer
• Thicker more expensive
• Relation between thickness and life is uncertain.
• Therefore, the total cost of the project is
  uncertain.

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Examples

• Design of an offshore drilling tower
• How safe is safe enough?
• Possibility of hurricane during useful life

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• Design of an off-shore wind turbine
• fatigue life is unknown
• must design to tradeoff initial costs with
  lifetime and reliability

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• Geotechnical design
• capacity of in situ subsoil and or rock deposit
  must be estimated, but will be uncertain
• How much margin of safety do you design in?

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Key points this semester

• How much is enough?
• How much testing?
• How safe is safe?
• When faced with evidence, what should you
  believe?
• When do we need to consider a distribution and
  not just an average? How do we propagate
  uncertainty from the input to the output?

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Preview Designing Experimentscontext and
controls

• A key question in all scientific explorations is
  compared to what?
• Having a control is essential for interpreting
  the results of a study.
• A controlled study is one of the following
• A study containing a control group
• A study in which the investigator assigns
  treatment and non-treatment (control).

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Designing ExperimentsRandomized versus
Observational

• In a randomized study the researcher assigns
  treatment randomly.
• In an observational study, the researcher
  observes differences between two groups
  treatment and control but does not assign
  treatment.

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Designing ExperimentsRandomized versus
Observational

• Observational studies can be very hard to
  interpret, since the subjects have often
  self-selected into the treatment and control
  groups.

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Observational Studies Example 1

• Pet a day keeps the doctor away
• Those who owned pets had contact with a doctor
  8.42 times in a year.
• Those who did not own pets had contact 9.49
  times.
• What interpretation is being made?
• Could there be another interpretation?

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Observational Studies Example 1

• Pet a day keeps the doctor away
• Those who owned pets had contact with a doctor
  8.42 times in a year.
• Those who did not own pets had contact 9.49
  times.
• Does Pet cause good health or
• Does good health cause pet or
• Does something else cause both?

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Newspaper article assignment

• Due Monday April 6.
• May want to collect article early.
• Find an article that uses statistics in a
  questionable way. Must be from a newspaper (not
  blog), dated 2009.
• Are they jumping from correlation to causation?
  Is there another reasonable interpretation? How
  might you test the two interpretations against
  one another?
• Was the sample carefully chosen?
• Were key things controlled for?
• Are they lacking context.

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Rest of Semester

• Probability Theory
• Random Variables
• Logic and mathematics of probability
• Probability Models
• Probability distributions that arise commonly
• binomial
• normal
• Allow us to represent real-world phenomena in a
  convenient way
• Statistical Inference
• How to make sense of a pile of data
• How to collect data sensibly
• How to determine what the data is really telling
  us

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Probability

• Probability is the mathematical language of
  uncertainty.

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Probability

• The theory of probability is used to provide
  methods for quantifying the chance, or
  likelihood, associated with different sets of
  outcomes.
• It is useful anytime we need to make a decision
  and we dont have perfect information.

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Examples

• The theory of probability is used to provide
  methods for quantifying the chance, or
  likelihood, associated with different sets of
  outcomes.
• Suppose each tile on the shuttle has a one in a
  million chance of failing.
• What is the probability that exactly two tiles
  fail?
• It does not matter which two
• What is the probability that these two particular
  tiles both fail?
• It does matter which two.
• What is the probability that at most two tiles
  fail?

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Basic Concepts

• Random experiment is the process of observing the
  outcome of a chance phenomenon
• The roll of a die
• Elementary outcome is any possible result of the
  random experiment
• The number on the side facing up.
• Sample space, S, is the collection of all
  possible elementary outcomes of the random
  phenomenon
• 1,2,3,4,5,6
• Event A is some collection of elementary outcomes
• An event A occurs when any of its elementary
  outcomes occur
• An event may be to roll an odd number 1,3,5

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Basic Concepts

• Events may be simple or compound
• Simple events
• cannot be broken into other events
• Single elementary outcome
• Compound events
• Made up of two or more simple events
• I.e., several elementary outcomes

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Examples Identify simple and compound events

• Random experiment toss two coins
• Sample space?
• Event exactly one head ? ?
• Random experiment Throw two dice
• Sample space?
• Event Sum of the two equals 6 ? ?

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Examples

• Random experiment Lifetime of an item
• Sample space ?
• Event the item lasts at least 1000 hours.
• Random experiment Number of good parts until one
  defective is produced
• Sample space ?
• Event A defective happens in the first 10
  production units

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Example

• A contractor is planning the purchase of
  bulldozers for a project in a remote area. He
  figures that there is a 50 chance each dozer can
  last at least 6 months. If he purchases 3, what
  is probability that there will be only 1 left in
  6 months?
• What is the appropriate sample space?

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Example

• A contractor is planning the purchase of
  bulldozers for a project in a remote area. He
  figures that there is a 50 chance each dozer can
  last at least 6 months. If he purchases 3, what
  is probability that there will be only 1 left in
  6 months?
• What is the appropriate sample space?
• Each dozer will either be working (W) or broken
  (B)
• WWW
• BWW
• WBW
• BBW
• WWB
• WBB
• BWB
• BBB

8 simple events in the sample space. Which simple
events make up the event of interest?
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Example

• A contractor is planning the purchase of
  bulldozers for a project in a remote area. He
  figures that there is a 50 chance each dozer can
  last at least 6 months. If he purchases 3, what
  is probability that there will be only 1 left in
  6 months?
• What is the appropriate sample space?
• Each dozer will either be working (W) or broken
  (B)
• WWW
• BWW
• WBW
• BBW
• WWB
• WBB
• BWB
• BBB

8 simple events in the sample space. Which simple
events make up the event of interest? BBW, WBB,
BWB prob is 3/8
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Set Theory

• The set of all possibilities in a probabilistic
  problem is the sample space.
• Sample spaces may be discrete or continuous
• If discrete, members of the space are countable
• Bulldozer example has a discrete sample space
• Lifetime example has continuous sample space

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Set Theory

• The set of all possibilities in a probabilistic
  problem is the sample space.
• Each member or sample point of the sample space
  is a simple event.
• Sample spaces may be discrete or continuous
• An event is a subset of the sample space.
• It contains one or more sample points.
• The realization of any of these sample points
  constitutes the occurrence of the event.

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Special Events

• Impossible event. This is an event with no
  members, an empty set in a sample space
• Certain event. This is denoted S, it contains all
  the sample points in the sample space.
• Complementary event For an event E in a
  sample space S, the complementary event contains
  all the sample points in S that are not in E.

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Venn Diagrams

• Complementary event

E
Sample space S
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Venn Diagrams

• Two events E1 and E2

E2
E1
Sample space S
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Venn Diagrams

• E1 one bulldozer
• E2 two bulldozers

bww wwb wbw
bbw bwb wbb
E2
www bbb
Sample space S
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Venn Diagrams
Throw one dice

• E1 dice is even
• E2 dice is less than 4

1,3
4, 6
5
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Sample space S