Ucertaty, Data, ad Judgmet Sesso 06 Structure of the Course Topc Sesso Probablty -5 Estmato 6-8 Hypothess Testg 9-10 Regresso 11-16 1
Mcrosoft AND Itel (50-50) You vest $,500 MSFT ad $,500 INTC X = Aual retur o $1 vested Mcrosoft: X ~ N(1.1,0.1 ) Y = Aual retur o $1 vested Itel : Y ~ N(1.1,0.1 ) The your et proft s W = -5,000 +,500 X +,500 Y Questos: What s your average et proft? What s the varace of your et proft? How lkely are you to make postve et proft? MSFT vs. INTC weekly
Correlato Plots Facts About Covarace Idcates the drecto of a relatoshp If COV[X,Y] < 0 X ad Y are versely related If COV[X,Y] > 0 X ad Y are postvely related If X ad Y are depedet COV[X,Y] = 0 3
Facts About Correlato Ut-free measure of the stregth of a lear relatoshp Les betwee -1 ad 1. If X ad Y are depedet, the CORR[X,Y] = 0. CORR[X,Y] COV[X,Y] X Y Correlato vs. Causato 4
Gettg Dzzy. W = a + b X + c Y E(W) = a + be(x) + ce(y) Var(W) = b Var(X) + c Var(Y) + bc[cov(x, Y)] Mcrosoft AND Itel (50-50) You vest $,500 MSFT ad $,500 INTC X ~ N(1.1,0.1 ), Y ~ N(1.1,0.1 ) Your et proft s W = -5,000 +,500 X +,500 Y W ~ N( W, w ) W = E(W)= -5,000 +,500 X +,500 Y = 500 W = Var(W) =,500 X +,500 Y + *,500*,500 * COV (X,Y) =,500 X +,500 Y + *,500*,500 * X Y CORR(X,Y) 5
INVESTING IN MICROSOFT AND/OR INTEL Sx Cases Case Acto Correlato E(W) St.Dev(W) 1 All Mcrosoft N/A ½ Mcrosoft + ½ Itel 0 3 ½ Mcrosoft + ½ Itel 0.6 4 ½ Mcrosoft + ½ Itel -0.6 5 ½ Mcrosoft + ½ Itel 1 6 ½ Mcrosoft + ½ Itel -1 Ivestmet: Dstrbuto of et profts Correlato E(W) St.Dev(W) 0 500 353.55 0.6 500 447.1-0.6 500 3.61 1 500 500-1 500 0 6
Fuctos of Radom Varables Today s materal s relevat to several mportat practcal problems: Moder Portfolo Theory Rsk Maagemet Reveue Plag Producto Plag Ad we oly scratched the surface. Key Questos UDJ 1. How to look at Data? (1-3) Judgmet ssues Measures of locato & dsperso Measures of extremes (Chebyshev s T., Emprcal Rule) How to model Ucertaty? (4-5) Dstrbutos: Bomal, Posso, Normal Fuctos of Radom Varables. How to make Estmatos? (6-8) Make educated guesses about populato parameters Say how cofdet you are those guesses 3. How to make Decsos based o data? (9-10) Hypothess testg / Correct Observatos/Theores 4. How to make Predctos based o data? (11-16) Regresso / forecastg 7
Some practcal questos What s the average salary of a INSEAD MBA after 5 years? What s the expected market sze for the Pad/a ew razor? What s the proporto of defectve cars produced by Toyota? What s the average rate of arrvals at a bak/hosptal? What percetage of techology startups go bust? What s the average watg tme for a tax? What s the average umber of vewers of a TV ad? What s the uemploymet rate the US? What s the average alcohol cosumpto at a INSEAD party? Populato x 1, x,..., x N Radom sample of small sze Parameters = Populato mea = Populato stadard devato Estmates Sample x 1,x,...,x X = Sample mea s = Sample stadard devato 8
Our Goal 1. Estmate the populato mea usg X. Quatfy the possble error we make usg X as a estmate of Mea ad Varace Populato Sample Mea Varace SD N x x x x x N s x 1 x x x N s s 1 9
What s the Populato Hstogram? The Cetral Lmt Theorem* For a radom sample of sze 30 take from a populato wth mea ad stadard devato : The sample mea X ~ Normal hstogram X = X = / * Ths holds for ay populato hstogram of X. s use as a estmate of If we do t kow sgma, the IF 30 10
Key Questo What s the max. possble error we make? Buldg a cofdece terval A Illustrato of the dervato of 95% cofdece tervals. 95% 1. 96 1. 96 X 3 0 3 Z 1.96 11
95% Cofdece Itervals From the dagram, we ca wrte: P 1.96 X 1.96 0.95 Algebra P X 1.96 X 1.96 0.95 Example The WSJ reports that from a study of 300 subscrbers they fd that the average come s 80.000 ad the std s 15.000. How close to that would the real average (of ALL readers) be wth 95% probablty/cofdece? 1
A Illustrato of the dervato of 95% cofdece tervals. 95% 1. 96 1. 96 X 3 0 3 Z 1.96 100(1-a)% Za Za X 0 Z Z a/ 13
100(1-a)% Cofdece Itervals From the dagram, we ca wrte: P Z X Z 1 a a a Algebra P X Z X Z 1 a a a Case 1: s kow, 30 X - / ~ Z Case : s ukow, 30 X - s/ ~ Z Cofdece Iterval Lmts X Z a X Z a s 14
Example: 99% Cofdece Aother way If you sample 100 subscrbers, what s the probablty that the error of your estmated average salary from these 100 subscrbers s less tha 1000? 15
Aother way The std of the come of ALL subscrbers to the WSJ s 15.000 How may subscrbers should we sample to be 95% cofdet that the average come of ALL the subscrbers s wth X of the sample mea? 16