Chapter 5 Student Lecture Notes 5-1

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1 Chapter 5 Student Lecture Notes 5-1 Basc Busness Statstcs (9 th Edton) Chapter 5 Some Important Dscrete Probablty Dstrbutons 004 Prentce-Hall, Inc. Chap 5-1 Chapter Topcs The Probablty Dstrbuton of a Dscrete Random Varable Covarance and Its Applcatons n Fnance Bnomal Dstrbuton Hypergeometrc Dstrbuton Posson Dstrbuton 004 Prentce-Hall, Inc. Chap 5- Random Varable Random Varable Outcomes of an experment expressed numercally E.g., Toss a de twce; count the number of tmes the number 4 appears (0, 1 or tmes) E.g., Toss a con; assgn $10 to head and -$30 to a tal = $10 T = -$ Prentce-Hall, Inc. Chap 5-3

2 Chapter 5 Student Lecture Notes 5- Dscrete Random Varable Dscrete Random Varable Obtaned by countng (0, 1,, 3, etc.) Usually a fnte number of dfferent values E.g., Toss a con 5 tmes; count the number of tals (0, 1,, 3, 4, or 5 tmes) 004 Prentce-Hall, Inc. Chap 5-4 Dscrete Probablty Dstrbuton Example Event: Toss Cons Count # Tals T T Probablty Dstrbuton Values Probablty 0 1/4 =.5 1 /4 =.50 1/4 =.5 T T Ths s usng the A Pror Classcal Probablty approach. 004 Prentce-Hall, Inc. Chap 5-5 Dscrete Probablty Dstrbuton Lst of All Possble [ j, P( j ) ] Pars j = Value of random varable P( j ) = Probablty assocated wth value Mutually Exclusve (Nothng n Common) Collectve Exhaustve (Nothng Left Out) ( j) P( j) 0 P 1 = Prentce-Hall, Inc. Chap 5-6

3 Chapter 5 Student Lecture Notes 5-3 Summary Measures Expected Value (The Mean) Weghted average of the probablty dstrbuton ( ) jp( j) µ = E = j E.g., Toss cons, count the number of tals, compute expected value: µ = j j ( j) P ( )( ) ( )( ) ( )( ) = = Prentce-Hall, Inc. Chap 5-7 Summary Measures (contnued) Varance Weghted average squared devaton about the mean σ = E ( µ ) = ( µ ) P( ) E.g., Toss cons, count number of tals, compute varance: ( ) P( ) σ = µ ( 0 1 ) (.5) ( 1 1 ) (.5) ( 1 ) (.5) = + + = Prentce-Hall, Inc. Chap 5-8 σ Covarance and Its Applcaton N = E Y E Y P Y ( ) ( ) ( ) ( ) Y = 1 : dscrete random varable th : outcome of Y : dscrete random varable Y Y th : outcome of P Y : probablty of occurrence of the outcome of and the 004 Prentce-Hall, Inc. Chap 5-9 th th outcome of Y

4 Chapter 5 Student Lecture Notes 5-4 Computng the Mean for Investment Returns Return per $1,000 for two types of nvestments Investment P( ) P(Y ) Economc Condton Dow Jones Fund Growth Stock Y.. Recesson -$100 -$ Stable Economy Expandng Economy ( ) ( )( ) ( )( ) ( )( ) E = µ = = $105 ( ) ( )( ) ( )( ) ( )( ) E Y = µ Y = = $ Prentce-Hall, Inc. Chap 5-10 Computng the Varance for Investment Returns Investment P( ) P(Y ) Economc Condton Dow Jones Fund Growth Stock Y.. Recesson -$100 -$ Stable Economy Expandng Economy σ (.)( ) (.5)( ) (.3)( ) = + + = 14,75 σ = Y (.)( ) (.5)( ) (.3)( ) 004 Prentce-Hall, Inc. Chap 5-11 σ = + + = 37,900 σ = Y Computng the Covarance for Investment Returns Investment P( Y ) Economc Condton Dow Jones Fund Growth Stock Y. Recesson -$100 -$00.5 Stable Economy Expandng Economy ( )( )(.) ( )( )(.5) + ( )( )(.3) = 3, 300 σ = + Y The covarance of 3,000 ndcates that the two nvestments are postvely related and wll vary together n the same drecton. 004 Prentce-Hall, Inc. Chap 5-1

5 Chapter 5 Student Lecture Notes 5-5 Computng the Coeffcent of Varaton for Investment Returns σ CV ( ) = = = 1.16 = 116% µ 105 σ Y CV ( Y ) = = =.16 = 16% µ Y 90 Investment appears to have a lower rsk (varaton) per unt of average payoff (return) than nvestment Y Investment appears to have a hgher average payoff (return) per unt of varaton (rsk) than nvestment Y 004 Prentce-Hall, Inc. Chap 5-13 Sum of Two Random Varables The expected value of the sum s equal to the sum of the expected values E( + Y) = E( ) + E( Y) The varance of the sum s equal to the sum of the varances plus twce the covarance Var ( + Y ) = σ + Y= σ + σy+ σ Y The standard devaton s the square root of the varance σ = σ + Y + Y 004 Prentce-Hall, Inc. Chap 5-14 Portfolo Expected Return and Rsk The portfolo expected return for a two-asset nvestment s equal to the weghted average of the two assets E( P) = we( ) + ( 1 w) E( Y) where w= porton of the portfolo value assgned to asset Portfolo rsk ( ) σ = w σ + 1 w σ + w( 1 w) σ P Y Y 004 Prentce-Hall, Inc. Chap 5-15

6 Chapter 5 Student Lecture Notes 5-6 Computng the Expected Return and Rsk of the Portfolo Investment Investment P( Y ) Economc Condton Dow Jones Fund Growth Stock Y. Recesson -$100 -$00.5 Stable Economy Expandng Economy Suppose a portfolo conssts of an equal nvestment n each of and Y: E( P ) = 0.5( 105) + 0.5( 90) = 97.5 ( ) ( ) ( ) ( ) ( )( )( ) σ = = P 004 Prentce-Hall, Inc. Chap 5-16 Usng PHStat PHStat Decson Makng Covarance and Portfolo Analyss Fll n the Number of Outcomes: Check the Portfolo Management Analyss box Fll n the probabltes and outcomes for nvestment and Y Manually compute the CV usng the formula n the prevous slde Here s the Excel spreadsheet that contans the results of the prevous nvestment example: Mcrosoft Excel Worksheet 004 Prentce-Hall, Inc. Chap 5-17 Important Dscrete Probablty Dstrbutons Dscrete Probablty Dstrbutons Bnomal Hypergeometrc Posson 004 Prentce-Hall, Inc. Chap 5-18

7 Chapter 5 Student Lecture Notes 5-7 Bnomal Probablty Dstrbuton n Identcal Trals E.g., 15 tosses of a con; 10 lght bulbs taken from a warehouse Mutually Exclusve Outcomes on Each Tral E.g., Heads or tals n each toss of a con; defectve or not defectve lght bulb Trals are Independent The outcome of one tral does not affect the outcome of the other 004 Prentce-Hall, Inc. Chap 5-19 Bnomal Probablty Dstrbuton Constant Probablty for Each Tral (contnued) E.g., Probablty of gettng a tal s the same each tme we toss the con Samplng Methods Infnte populaton wthout replacement Fnte populaton wth replacement 004 Prentce-Hall, Inc. Chap 5-0 Bnomal Probablty Dstrbuton Functon n! n P( ) = p ( 1 p)! ( n )! P( ) : probablty of successes gven n and p : number of "successes" n sample 0,1,, n p : the probablty of each "success" n : sample sze Tals n Tosses of Con P() 0 1/4 =.5 1 /4 =.50 1/4 =.5 ( = L ) 004 Prentce-Hall, Inc. Chap 5-1

8 Chapter 5 Student Lecture Notes 5-8 Bnomal Dstrbuton Characterstcs Mean µ = E ( ) = np E.g., µ = np = 5.1 ( ) =.5 Varance and Standard Devaton σ = np 1 p E.g., ( ) ( 1 p) σ = np P() ( p) ( )( ) n = 5 p = σ = np 1 = = Prentce-Hall, Inc. Chap 5- Bnomal Dstrbuton n PHStat PHStat Probablty & Prob. Dstrbutons Bnomal Example n Excel Spreadsheet Mcrosoft Excel Worksheet 004 Prentce-Hall, Inc. Chap 5-3 Example: Bnomal Dstrbuton A md-term exam has 30 multple choce questons, each wth 5 possble answers. What s the probablty of randomly guessng the answer for each queston and passng the exam (.e., havng guessed at least 18 questons correctly)? Are the assumptons for the bnomal dstrbuton met? Yes, the assumptons are met. Usng results from PHStat: n = 30 p = 0. Mcrosoft Excel Worksheet P 18 = ( ) ( ) Prentce-Hall, Inc. Chap 5-4

9 Chapter 5 Student Lecture Notes 5-9 Hypergeometrc Dstrbuton n Trals n a Sample Taken from a Fnte Populaton of Sze N Sample Taken Wthout Replacement Trals are Dependent Concerned wth Fndng the Probablty of Successes n the Sample Where There are A Successes n the Populaton 004 Prentce-Hall, Inc. Chap 5-5 Hypergeometrc Dstrbuton Functon A N A E.g., 3 Lght bulbs were n selected from 10. Of the 10, P( ) = there were 4 defectve. What N s the probablty that of the n 3 selected are defectve? P ( ): probablty that successes gven n, N, and A n : sample sze 4 6 N : populaton sze 1 P ( ) = =.30 A : number of "successes" n populaton 10 3 : number of "successes" n sample = 0,1,, L, n ( ) 004 Prentce-Hall, Inc. Chap 5-6 Hypergeometrc Dstrbuton Characterstcs Mean A µ = E ( ) = n N Varance and Standard Devaton σ ( ) na N A N n = N N 1 na( N A) N n σ = N N 1 Fnte Populaton Correcton Factor 004 Prentce-Hall, Inc. Chap 5-7

10 Chapter 5 Student Lecture Notes 5-10 Hypergeometrc Dstrbuton n PHStat PHStat Probablty & Prob. Dstrbutons Hypergeometrc Example n Excel Spreadsheet Mcrosoft Excel Worksheet 004 Prentce-Hall, Inc. Chap 5-8 Posson Dstrbuton Sméon Posson 004 Prentce-Hall, Inc. Chap 5-9 Posson Dstrbuton Dscrete events ( successes ) occurrng n a gven area of opportunty ( nterval ) Interval can be tme, length, surface area, etc. The probablty of a success n a gven nterval s the same for all the ntervals The number of successes n one nterval s ndependent of the number of successes n other ntervals The probablty of two or more successes occurrng n an nterval approaches zero as the nterval becomes smaller E.g., # customers arrvng n 15 mnutes E.g., # defects per case of lght bulbs 004 Prentce-Hall, Inc. Chap 5-30

11 Chapter 5 Student Lecture Notes 5-11 Posson Probablty Dstrbuton Functon e ( ) λ λ P =! P( ) : probablty of "successes" gven λ : number of "successes" per unt λ : expected (average) number of "successes" e :.7188 (base of natural logs) E.g., Fnd the probablty of e 3.6 customers arrvng n 3 mnutes P( ) = = 4! when the mean s Prentce-Hall, Inc. Chap 5-31 Posson Dstrbuton n PHStat PHStat Probablty & Prob. Dstrbutons Posson Example n Excel Spreadsheet Mcrosoft Excel Worksheet P ( = x λ e -λ x λ x! 004 Prentce-Hall, Inc. Chap 5-3 Mean µ = E( ) = λ N = P = 1 Posson Dstrbuton Characterstcs ( ) Standard Devaton and Varance σ = λ σ = λ P() P() λ = λ = Prentce-Hall, Inc. Chap 5-33

12 Chapter 5 Student Lecture Notes 5-1 Chapter Summary Addressed the Probablty Dstrbuton of a Dscrete Random Varable Defned Covarance and Dscussed Its Applcaton n Fnance Dscussed Bnomal Dstrbuton Addressed Hypergeometrc Dstrbuton Dscussed Posson Dstrbuton 004 Prentce-Hall, Inc. Chap 5-34