QM 2113 - Spring 2002QM 2113 - Spring 2002

Business StatisticsBusiness Statistics

Probability Probability DistributionsDistributions

Student ObjectivesStudent Objectives

Discuss homework solutions (review of Discuss homework solutions (review of material from previous classmaterial from previous class– Compute simple probabilities

• Joint (with independent events)• Conditional• Union (of mutually exclusive events)

– Discuss the concept of independence with respect to probability

Define Define probability distributionprobability distribution Calculate and use summary measures of Calculate and use summary measures of

probability distributions for simple probability distributions for simple decision analysisdecision analysis

Recall Relative Recall Relative FrequencyFrequency

Regardless of method used to Regardless of method used to determine probability, it can be determine probability, it can be interpreted as relative frequencyinterpreted as relative frequency– Recall that relative frequency is observed

proportion of time some event has occurred• Sites developed in-house• Incomes between $10,000 and $20,000

– Probability is just expected proportion of time we expect something to happen in the future given similar circumstances

Note also, proportions are probabilitiesNote also, proportions are probabilities Exercise: Worst National BankExercise: Worst National Bank

WNB: The DataWNB: The Data

Employee X 1 X 2 X 6 X 7 X 8

1 21.0 38 1 14 02 38.3 41 2 5 13 75.0 40 5 12 0216 57.2 62 4 25 0217 37.2 51 2 18 0218 34.2 53 2 17 1219 34.7 40 2 10 1220 35.2 51 2 5 1221 74.9 49 5 8 0

Variables are: Employee Employee number (1 to 221)

X 1 Annual salary, in thousands of $

X 5 Using computer = 1, not using = 0

X 6 Job classification: 1 = lowest, 6 = highest

X 7 Experience at this bank (in years)

X 8 Male = 0, female = 1

WNB: A SummaryWNB: A Summary

Job A (Male) B (Female)F (Level 1) 9.05% 14.48% 23.53%G (Level 2) 8.14% 9.95% 18.09%H (Level 3) 5.43% 9.50% 14.93%I (Level 4) 6.79% 11.31% 18.10%J (Level 5) 5.88% 10.41% 16.29%K (Level 6) 2.71% 6.33% 9.04%

38.00% 61.98%

Gender

Looking at the Job Looking at the Job VariableVariable

WNB Job Category Breakdown

F (Level 1)24%

G (Level 2)18%

H (Level 3)15%

I (Level 4)18%

J (Level 5)16%

K (Level 6)9%

Now, the Gender Now, the Gender VariableVariableWNB Gender Breakdown

A (Male)38%

B (Female)62%

For Event J (Level 5), For Event J (Level 5), a Look at Gendera Look at Gender

Gender Breakdown for Level 5

A (Male)36%

B (Female)64%

Consider Job vs Consider Job vs GenderGender

Some proportions (probabilities)Some proportions (probabilities)– First: P(B) = ???– How about: P(J)?– Now: P(B and J) = ???– What about: P(H or J)?– Finally: P(B | J) = ???

Think about what these results Think about what these results representrepresent

Now, does P(B) have anything to Now, does P(B) have anything to do with J?do with J?

For decision problems that occur more For decision problems that occur more than once, we can often estimate than once, we can often estimate probabilities from historical data. probabilities from historical data.

Other decision problems represent one-Other decision problems represent one-time decisions where historical data for time decisions where historical data for estimating probabilities don’t exist. estimating probabilities don’t 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 expert’s opinions.

We will focus on techniques that can be We will focus on techniques that can be used once appropriate probability used once appropriate probability estimates have been obtained.estimates have been obtained.

Probability Probability CommentsComments

Probability: Not Just Probability: Not Just for Theoreticiansfor Theoreticians

Typical probability Typical probability applicationsapplications– Statistical inference– Decision analysis– Reliability

But first, we need to know But first, we need to know something about something about distributionsdistributions

So, What’s a So, What’s a Distribution?Distribution?

Applies to random variablesApplies to random variables– 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

Distribution for a random variableDistribution for a random variable– Exhaustive list of mutually exclusive events– Corresponding probabilities– Essentially a relative frequency distribution– Note: probabilities sum to 1.00

Consider an Consider an InvestmentInvestment

EPS is not certain!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 So many numbers; how do we decide whether or not to invest?decide whether or not to invest?

Summarize!Summarize!– Expected value (i.e., the average)– Variance (or standard deviation)

The DistributionThe Distribution

The info:The info:

OK, what number best summarizes OK, what number best summarizes EPS?EPS?

x P(x)-20 5%-10 10%0 30%10 20%20 20%30 10%40 5%

Summarizing Summarizing Random VariablesRandom Variables

Much like summarizing observed Much like summarizing observed variables (quantitative)variables (quantitative)– Central tendency– Variability

Expected valueExpected value– Certainty equivalent– E(x) = = xP(x) = weighted average!

Standard deviationStandard deviation– Summarizes expected (average) variation– = square root of (x- )2P(x)

An Alternative An Alternative InvestmentInvestment

The infoThe info

Now, let’s summarize this Now, let’s summarize this alternativealternative

x P(x)-20 16.7%-10 16.7%0 16.7%10 16.7%20 16.7%30 16.7%

Decision Analysis: A Decision Analysis: A Structured ApproachStructured Approach What decisions have What decisions have youyou made made

lately?lately?– Personal– Work related

Consider the decisions our Consider the decisions our national leaders have made national leaders have made latelylately

Let’s address what they have Let’s address what they have in commonin common

Consider the Consider the Scientific Problem Scientific Problem Solving ProcessSolving Process

Define problem:Define problem:– What do we control?– What’s important?– Other . . .

Identify alternativesIdentify alternatives Evaluate alternativesEvaluate alternatives Select “best” alternativeSelect “best” alternative Implement solutionImplement solution Monitor processMonitor process

Now, this very nearly summarizesNow, this very nearly summarizesdecision analysisdecision analysis

Consider Two Consider Two Aspects of Any Aspects of Any

DecisionDecision Courses of actionCourses of action

– What choices we have– Examples: which job, how many workers, .

. .

States of natureStates of nature– Events out of our control– Examples: who’s elected, weather, court

decision (Microsoft), economy– Described by probability distribution

So, What Do We Do So, What Do We Do With It?With It?

Use it to choose courses of actionUse it to choose courses of action Determine Determine certainty equivalencecertainty equivalence

– Gives us a single number– This is the expected value

ExamplesExamples– Investments– Product purchases– Others . . .

General Procedure General Procedure for Decision Analysisfor Decision Analysis Determine alternativesDetermine alternatives For each alternativeFor each alternative

– Determine outcomes (e.g., monetary values) possible

– Determine probabilities for those outcomes Create model (matrix or tree)Create model (matrix or tree) Determine EMV for each alternativeDetermine EMV for each alternative Make choiceMake choice

– Best EMV?– Consider risk

PProbability robability Distributions:Distributions:

A Broader ViewA Broader View Random variablesRandom variables

– Discrete– Continuous

Normal distributions, the most Normal distributions, the most well known continuous well known continuous distributiondistribution

Tune in next time for more . . .Tune in next time for more . . .