Uncertainty, Data, and Judgment

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1 Ucertaty, Data, ad Judgmet Sesso 06 Structure of the Course Topc Sesso Probablty -5 Estmato 6-8 Hypothess Testg 9-10 Regresso

2 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

3 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

4 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

5 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

6 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 ½ Mcrosoft + ½ Itel ½ Mcrosoft + ½ Itel 1 6 ½ Mcrosoft + ½ Itel -1 Ivestmet: Dstrbuto of et profts Correlato E(W) St.Dev(W)

7 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

8 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

9 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

10 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

11 Key Questo What s the max. possble error we make? Buldg a cofdece terval A Illustrato of the dervato of 95% cofdece tervals. 95% X Z

12 95% Cofdece Itervals From the dagram, we ca wrte: P 1.96 X Algebra P X 1.96 X Example The WSJ reports that from a study of 300 subscrbers they fd that the average come s ad the std s How close to that would the real average (of ALL readers) be wth 95% probablty/cofdece? 1

13 A Illustrato of the dervato of 95% cofdece tervals. 95% X Z (1-a)% Za Za X 0 Z Z a/ 13

14 100(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

15 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

16 Aother way The std of the come of ALL subscrbers to the WSJ s 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

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot