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ENMA 420/520 Statistical Processes Spring 2007. Michael F. Cochrane, Ph.D. Dept. of Engineering Management Old Dominion University. Class Five Readings & Problems. Reading assignment M & S Chapter 6 Sections 6.1 - 6.8 Recommended problems M & S Chapter 6
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ENMA 420/520Statistical ProcessesSpring 2007 Michael F. Cochrane, Ph.D. Dept. of Engineering Management Old Dominion University
Class FiveReadings & Problems • Reading assignment • M & S • Chapter 6 Sections 6.1 - 6.8 • Recommended problems • M & S Chapter 6 • Chapter 6: 5, 13, 18, 20, 29, 38, 40
Chapter 6Bivariate Distributions • What do we mean by univariate probability distributions? • What do we mean by multivariate probability distributions? • Special case of multivariate probability distributions: Bivariate distributions
Recall Bivariate Distributions • Looking for p(2 things happening) • p(XY) • Alternative way of writing p(xy) • p(X,Y) • What does p(x, y) really mean? • p(X = x0, Y = y0)
x P(x,y) y Example Bivariate DistributionToss 2 Dice What is p(1, 1)?
How do you interpret? Over y Discrete rv Continuous rv GeneralizingMarginal Probabilities • Generalize from previous example:
Bivariate DistributionsExtending the Univariate Case Note the extensions from the univariate special case!
Very useful relationships Recall: E(.) more generally a linear operator Bivariate DistributionsWhat if x & y Are Independent? Do not forget: above only valid if x & y are independent! Note the convenience of assuming independence!
Point (xi, yj) CovarianceLooking for Relationship Between x & y Does knowing x tells us anything about y?
CovarianceLooking for Relationship Between x & y Does knowing x tell you something about corresponding y? Covariance a measure of strength of relationship!
CovarianceHow Do x & y Vary? Recall VAR(y): VAR(y)=E(y - y)2 = y2 Covariance COV(x,y) = E[(x - x) (y - y)] =xy2 What would you expect COV here to be?
CovarianceHow Do x & y Vary? What is (x - x) (y - y) for these points? y What is (x - x) (y - y) for these points? x Covariance COV(x,y) = E[(x - x) (y - y)] =xy2 What can you say about COV for a set of points like these?
Covariance of Independent Variables • Recall formula for COV(x,y) COV(x,y) = E[(x - x) (y - y)] = E(xy) - xy Make sure you can derive the above What happens if x & y are independent? COV(x,y) = 0 Why?? Suppose COV(x,y) = 0, are x and y independent??
Covariance of Independent VariablesA One Way Street • If x & y are independent • Then COV(x,y)=0 • If COV(x,y) = 0 • Then cannot conclude x & y are independent
CovarianceA Unit Dependent Measure • Suppose x = height in feet and y = weight in pounds What are the units of COV(x,y)? Same observations: x now cm & y now g, what happens to COV(x,y)? • How did we address same issues in univariate case CV = / • Bivariate counterpart: Coefficient of Correlation What are the units? What is range of values?
Range of possible values: -1 1 Here = 1 All observations on a line But here also = 1 What does = 1 imply? What about the value of slope? Coefficient of CorrelationPossibilities What does = -1 mean? How about = 0?
Bivariate RelationshipsLinear Functions of RVs • Recall simple example from MBA 101: P = S - C • If S & C are both rv P is also a rv • Is P a linear function of S & C? • What is E(P)? • What is VAR(P)? • What is the distribution of P? • Easy to tell only for special cases • For example if S ~ N AND C ~ N • Lets generalize for linear functions of rv
Note the pattern Mean & VarianceLinear Functions of RVs • Define l l = a1y1 + … + anyn ai = constants yj = random variables l =linear function of n random variables • What is the E( l )? • VAR(l) = VAR (a1y1 + … + anyn) = a1212 + … + an2n2 + 2a1a2COV(y1,y2) + 2a1a3COV(y1,y3) + … + 2a1anCOV(y1,yn) + … + 2an-1anCOV(yn-1,yn) What happens when all yi are independent random variables?
Short ExerciseSimple Example • Consider a nut & bolt r = bolt radius, a rv r = 0.5 inches; r = 0.01 inches w = nut width, a rv w = 0.51 inches; w = 0.001 inches • What are r & r2 in centimeters? • Nut is placed around bolt • Let gap = g = w - r • What are g & g ? r w
Functions of Random VariablesSetting the Stage for Inferential Statistics • Random variables are just a type of variable w = x + y • What type of variable is w? & What is its distribution? IF: • x=12 and y=3 (i.e., both constants) • x~N(12,4) and y=3 (i.e., x =rv & y=constant) • x~N(12,4) and y~N(3,1) (i.e., x & y both rv) • x~U(0,1) and y~U(0,1) • Will verify using @Risk
@RISKUseful Add-In to Excel • Basic concepts of @RISK software • Add-in to Excel • Builds on basic spreadsheet capabilities • Allows definition of random variables within cells • Excel only allows constants • Results determined using simulation techniques • Will review simple concepts of simulation
Spreadsheet Simulation Build spreadsheet Incorporate rv’s Define number of iterations in simulation Run simulation Gather data of interest Output results
Performing a SimulationDoing 1 Iteration Do these steps n times during a single run of a simulation. Sample each rv in spreadsheet model Illustrate based on previously questions regarding w = x + y After simulation have n values (observations) for each output value of interest. What do these n output values represent? Perform calculations in the model Save output values of interest
What is the difference? Population Sampling Issues • Sampling key to inferential statistics • Random sampling from • Population (actual members of set) • A distribution representing a population Set of n observations within sample Random sample RANDOM is key word!
Sampling From a Population • Methods to be discussed • Random number table • Excel sampling data analysis tool • Statistical analysis sw all have capability Sample Extract n observations Population
N - # in population n - # in sample Could use random # table or sw Generate n random integers in range [1, N] Associate random integers with members of population Choose appropriate observations Random Number TableSimple Approach
Using Table 1 of Appendix B(page 897 in M&S) Problem: Have population of 100 members want random sample of 5 observations • Approach • Number observations 1 ==> 100 • Go to any page in table & select 5 adjacent numbers • Use last 3 digits to designate selected observations • Question • What if there were 130 members in population? • How would you adapt the method in this case?
More Practical Approaches • Systematic sample • Select every kth element of population • Useful for very large populations, why? • What is implicit assumption of this method? • Software more practical approach • Excel • MiniTab or equivalent statistical analysis sw • What if you want random sample from a rv • Recall rv represents a “population” • Rv described by a probability distribution
Population y y2 What type of variables are these How big a sample would you need in order for these to be equal? Inferential StatisticsLooking for Insight into Population 1 Sample of n
Both of these distributions have a mean and a standard deviation. f(s) s Sampling Distribution Standard error of a statistic is standard deviation of its sampling distribution. Do you recall seeing it in Excel output?
Class FiveReadings & Problems • Reading assignment • M & S • Chapter 6 Sections 6.1 - 6.8 • Chapter 7 Sections 7.1 - 7.2 • Recommended problems • M & S Chapter 6 • Chapter 6: 5, 13, 18, 20, 29, 38, 40