Basic Probability and Stats Review Random variables Discrete probability distributions Expected value of a discrete random variable Continuous Distributions Working with distributions in Excel

Random variables (RV) and probability distributions RV is a variable whose value depends on the outcome of an uncertain event(s) Low bid by competing firms, project completion date Demand for some product or service next year # of patients requiring open heart surgery next month at Hospital H Cost of Drug X in December, 2004 Probability of various outcomes determined by probability distribution associated with the RV Probability distributions are the “shapes of RV’s” As modelers, we select appropriate distributions Probability distributions mathematical functions Assign numeric probabilities to uncertain events modeled by the distribution See “Distributions, Simulation and Excel Functions” handout that Prof. Doane created and that I’ve posted on Web.

Two Types of Variables Discrete Distributions Integer, countable X example: # of warranty claims in a day P(X) is the probability at each point P(X) may be summed over X values Continuous Distributions X defined over an interval example: Length of stay for open heart surgery patients Points have no area Calculus gives area under curve

Discrete RVs and Probability Distributions DistributionReview.xls Countable # of outcome values Each possible outcome has an associated probability Expected Value of Discrete RV A few discrete distributions Empirical Binomial – BINOMDIST() Poisson – POISSON() Expected Demand Total Probability

Continuous RVs and Probability Distributions Infinite # of outcome values Has a probability distribution function (pdf), f(x), which you can loosely think of as P[X=b]. We calculate probabilities over intervals using the cumulative distribution function (cdf), F(x), which is P[X<=b] Area under the f(x) curve from –infinity to b Uniform f(x) Exponential f(x) Normal f(x)

P.D.F. vs. Cumulative Probability Density Function X axis shows values of X Y axis shows probability S P(X) = 1 if discrete  f(x) = 1 if continuous Histogram is pdf for data Cumulative Distribution Function Y axis shows cumulative probability 0  F(X)  1 and is non-decreasing

RiskView Excel Add-In, Part of Palisade Decision Tools Suite “Live” distribution viewing, Huge number of distributions Online Help has background info on distributions Start | Palisade Decision Tools | RiskView 4.5 Can also launch from within Excel from the Palisade Decision Tools toolbar (which is visible if any of the Palisade tools are running, e.g. @Risk) Have all create a Uniform(01,1). Interpret table and sliders Change to Normal(0,1), Interpret tables and sliders

A few useful distributions

The Normal Distribution Two parameters: Mean, standard deviation Symmetric Standard normal distribution has mean=0, std dev=1 Normally distributed data with any mean and standard deviation can be converted to a N(0,1) by standardizing X~N(m,s) Z~N(0,1) Excel has a number of functions related to the normal distribution: NORMDIST(), NORMINV() NORMSDIST(), NORMSINV() Let’s review handout “Excel Functions for Working with Normal Distributions” and do the Continuous tab in DistributionReview.xls

Distribution Review Download DistributionReview.xls Let’s answer questions on sheet Discrete We’ll do Continuous sheet momentarily Excel has many probability and statistical related functions Remember, probability distributions are a type of model for some uncertain quantity Think of histograms as empirical probability distribution functions

Descriptive Statistics in Excel Data Analysis Tool-Pak AVERAGE(), STDEV(), MEDIAN() FREQUENCY() PERCENTILE() RANK(), PERCENTRANK() MIN(), MAX() 2 ways to create histograms Data Analysis Tool-Pak Default bins User specified bins FREQUENCY() array function StatReview.xls