QM Spring 2002 Business Statistics Probability Distributions

Student Objectives  Discuss homework solutions (review of material 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 probability distribution  Calculate and use summary measures of probability distributions for simple decision analysis

Recall Relative Frequency  Regardless of method used to determine probability, it can be interpreted 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 probabilities  Exercise: Worst National Bank

WNB: The Data

WNB: A Summary

Looking at the Job Variable

Now, the Gender Variable

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

Consider Job vs Gender  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 represent  Now, does P(B) have anything to do with J?

 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 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 used once appropriate probability estimates have been obtained. Probability Comments

Probability: Not Just for Theoreticians  Typical probability applications – Statistical inference – Decision analysis – Reliability  But first, we need to know something about distributions

So, What’s a Distribution?  Applies 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 variable – Exhaustive list of mutually exclusive events – Corresponding probabilities – Essentially a relative frequency distribution – Note: probabilities sum to 1.00

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)

The Distribution  The info:  OK, what number best summarizes EPS?

Summarizing Random Variables  Much like summarizing observed variables (quantitative) – Central tendency – Variability  Expected value – Certainty equivalent – E(x) =  =  xP(x) = weighted average!  Standard deviation – Summarizes expected (average) variation –  = square root of  (x-  ) 2 P(x)

An Alternative Investment  The info  Now, let’s summarize this alternative

Decision Analysis: A Structured Approach  What decisions have you made lately? – Personal – Work related  Consider the decisions our national leaders have made lately  Let’s address what they have in common

Consider the Scientific Problem Solving Process  Define problem: – What do we control? – What’s important? – Other...  Identify alternatives  Evaluate alternatives  Select “best” alternative  Implement solution  Monitor process Now, this very nearly summarizes decision analysis

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: who’s elected, weather, court decision (Microsoft), economy – Described by probability distribution

So, What Do We Do With It?  Use it to choose courses of action  Determine certainty equivalence – Gives us a single number – This is the expected value  Examples – Investments – Product purchases – Others...

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 EMV for each alternative  Make choice – Best EMV? – Consider risk

P robability Distributions: A Broader View  Random variables – Discrete – Continuous  Normal distributions, the most well known continuous distribution  Tune in next time for more...