Statistics for Decision Making

1
Statistics for Decision Making
QM 2113 -- Spring 2003

• Probability Distributions Decision Analysis

Instructor John Seydel, Ph.D.
2
Student Objectives

• Apply expected value and variance concepts to
  decision problems
• Define probability distribution
• Calculate and use summary measures of probability
  distributions for simple decision analysis
• Explain what a random variable is and provide
  some examples
• Distinguish between discrete and continuous
  random variables
• Determine probabilities for random variables that
  are normally distributed

3
Student Objectives

• Distinguish between discrete and continuous
  random variables
• Identify from data when a variable is normally
  distributed
• Discuss characteristics of all normally
  distributed random variables
• Relate probability for continuous variables to
  area under the curve
• Calculate probabilities associated with normally
  distributed random variables

4
Expected Value and Variance

• Applies to random variables
• A random variable is a rule that assigns numeric
  values to outcomes of events
• Examples
• Amount of bicycles purchased on a given day
• Profit expected for various economic conditions
• Time required to complete a sales transaction
• Probability distribution for a random variable
• Exhaustive list of mutually exclusive events
• Corresponding probabilities
• Essentially a relative frequency distribution
• Note probabilities sum to 1.00

5
Consider an Investment

• EPS is not certain!
• Possibly 10 per share
• But maybe 20 per share
• Could even be as high as 50 per share
• But could also be as low as -20 per share
• So many numbers how do we decide whether or not
  to invest?
• Summarize!
• Expected value (i.e., the average)
• Variance (or standard deviation)

6
The Distribution for This Example

• The info
• OK, what number best summarizes EPS?
• Sound familiar (remember first day of class)?

7
Summarizing Random Variables

• Much like summarizing observed variables
  (numeric)
• Central tendency
• Variability
• Expected value
• Certainty equivalent
• E(x) m SxP(x) weighted average!
• Standard deviation
• Summarizes expected (average) variation
• s square root of S(x- m)2P(x)

8
Now, Lets Apply These To The Info Given for This
Alternative

• Expected EPS SxP(x) . . . ?
• Variance S(x- m)2P(x) . . . ?
• Is this good or bad?
• Need to compare to another opportunity . . .

9
An Alternative Investment

• The info
• Now, lets summarize this alternative
• Expected EPS
• Variance

10
This Is Decision Analysis

• Simply comparing two or more alternative courses
  of action
• Incorporate
• Multiple possible outcomes for each alternative
• Probability distributions for those outcomes
• More formally . . .

11
Consider Two Aspects of Any Decision

• Courses of action
• What choices we have
• Examples which job, how many workers, . . .
• States of nature
• Events out of our control
• Examples whos elected, weather, court decision
  (Microsoft), economy
• Described by probability distribution

12
So, What Do We Do With This Information?

• Use it to choose best course of action
• Determine certainty equivalence for each
  alternative (course of action)
• Gives us a single number
• This is the expected value
• Examples
• Investments
• Product purchases
• Others . . .

13
General Procedure for Decision Analysis

• Determine alternatives
• For each alternative
• Determine outcomes (e.g., monetary values)
  possible
• Determine probabilities for those outcomes
• Create model (matrix or tree)
• Determine expected value for each alternative
• Make choice
• Best expected value?
• Consider risk (i.e., look at variance)

14
Now, a Broader View of Probability Distributions

• Random variables are either
• Discrete or
• Continuous
• Normal distributions, the most well known
  continuous distribution
• Lets take a closer look . . .

15
Continuous Random Variables

• Any number of values are possible between any two
  given values
• Typically measured rather than counted
• Limited practically by ability to measure
• Examples
• Time required to process request
• Earnings per share
• Distance traveled in a given amount of time
• Monetary values (e.g., annual family incomes)
• Now, how about examples of variables that are
  discrete (i.e., not continuous)?

16
Continuous Probability Distributions

• Technically, like discrete distributions
• List of possible values
• Corresponding likelihood for each value
• However, the number of possible values is
  infinite
• Hence its impossible to list as we do with
  discrete distributions
• Generally we use tables when working with
  continuous distributions

17
But Before We Go Any Further . . .

• Any distribution is either
• Empirical we work directly with the relative
  frequencies
• Or theoretical we can specify probabilities as
  mathematical functions
• P(x) lxe-l / x!
• P(x lt t) 1 - e-lt
• But we usually approximate using theoretical
  distributions
• Most well known normal distributions
• Specific mathematical formula
• Other distributions, other formulae

18
Now, For An Example

• You need a car that gets at least 30 mpg
• Suppose a particular model of car has been tested
• Average mpg 34
• Standard deviation 3 mpg
• Typically histograms for this type of thing look
  like
• That is, mpg is approximately normally distributed

19
If Somethings Normally Distributed

• Its described by
• m (the population/process average)
• s (the population/process standard deviation)
• Histogram is symmetric
• Thus no skew (average median)
• So P(x lt m) P(x gt m) . . . ?
• Shape of histogram can be described by
• f(x) (1/sv2p)e-(x-m)2/2s 2
• We determine probabilities based upon distance
  from the mean (i.e., the number of standard
  deviations)

20
Our Problem at Hand

• We need a car that gets at least 30 mpg
• How likely is it that this model of vehicle will
  meet our needs?
• That is, P(x gt 30) . . . ?
• First, sketch
• Number line with
• Average
• Also x value of concern
• Curve approximating histogram
• Identify areas of importance
• Then determine how many sigma 30 is from mu
• Now use the table
• Finally, put it all together

21
Comments on the Problem

• A sketch is essential!
• Use to identify regions of concern
• Enables putting together results of calculations,
  lookups, etc.
• Doesnt need to be perfect just needs to
  indicate relative positioning
• Make it large enough to work with needs
  annotation (probabilities, comments, etc.)
• Now, what do we do with the probability weve
  just determined?
• Make a decision!

22
Using the Normal Table

• The outside values are z-scores
• That is, how many standard deviations a given x
  value is from the average
• Use these values to look up probabilities
• The body of the table indicates probabilities
• Note This is not a z table!
• We can (and do) also work in reverse
• Given a probability, determine z
• Once we have z we can determine what x value
  corresponds to that probability

23
Some Other Exercises

• Let x N(34,3) as with the mpg problem
• Determine
• Tail probabilities
• F(30) which is the same as P(x 30)
• P(x gt 40)
• Tail complements
• P(x gt 30)
• P(x lt 40)
• Other
• P(32 lt x lt 33)
• P(30 lt x lt 35)
• P(20 lt x lt 30)

24
Keep In Mind

• Probability proportion of area under the normal
  curve
• What we get when we use tables is always the area
  between the mean and z standard deviations from
  the mean
• Because of symmetry
• P(x gt m) P(x lt m) 0.5000
• Tables show probabilities rounded to 4 decimal
  places
• If z lt -3.09 then probability 0.5000
• If z gt 3.09 then probability 0.5000
• Theoretically, P(x a) 0
• P(30 x 35) P(30 lt x lt 35)

25
Why Is This Important?

• Some practical applications
• Process capability analysis
• Decision analysis
• Optimization (e.g., ROP)
• Reliability studies
• Others
• Most importantly, the normal distribution is the
  basis for understanding statistical inference
• Hence, bear with this it should be apparent soon

26
Summary of Objectives
27
Homework

• Current
• Chi square test of independence
• Expected value and variance
• Other . . . ?
• For next time
• Read selected material from text
• Tests of independence
• Decision analysis exercises
• Normal distribution exercises

28
Appendix
29
Populations and Samples
Population
Sample
Statistic
Parameter
30
Consider the Scientific Problem Solving Process

• Define problem
• What do we control?
• Whats important?
• Other . . .
• Identify alternatives
• Evaluate alternatives
• Select best alternative
• Implement solution
• Monitor process
• Now, this very nearly summarizesdecision analysis

31
Probability Comments

• For decision problems that occur more than once,
  we can often estimate probabilities from
  historical data.
• Other decision problems represent one-time
  decisions where historical data for estimating
  probabilities dont exist.
• In these cases, probabilities are often assigned
  subjectively based on interviews with one or more
  domain experts.
• Highly structured interviewing techniques exist
  for soliciting probability estimates that are
  reasonably accurate and free of the unconscious
  biases that may impact an experts opinions.
• We will focus on techniques that can be used once
  appropriate probability estimates have been
  obtained.