arg02_quantitative tools and excel

7/30/2019 Arg02_Quantitative Tools and Excel

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Some Quantitative Tools and Their Use in Excel

FIMMOD - FInancial Management MODels (750562)

(University of Brescia) 1 / 42
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Main topics

1 Linear Programming

2 StatisticsDescriptive StatisticsLinear RegressionProbability

Inference



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Linear ProgrammingGeneral statements

Managers often have to make decisions on how best to allocate scarceresources among competing activities

Limited resources are used to manufacture various products, orprovide services

Typically, there are many dierent ways to produce these products orservices. Management's problem is to nd the best way, given thelimitations of the resources

Linear programming (LP) is a mathematical technique designed to

help in the planning and allocation of the resources

Both the objective function and the constraints of an LP problemmust be linear

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help in the planning and allocation of the resources

Both the objective function and the constraints of an LP problemmust be linear

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help in the planning and allocation of the resourcesBoth the objective function and the constraints of an LP problemmust be linear

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An LP problem consists in

an objective function, typically a prot to maximize or a cost tominimize

constraints due to the fact that scarce resources imply limitations tothe courses of actions available

A typical application of LP is the search for the mix of nancialinstruments driving to the highest possible return

Non linear programming involves an objective functionand/constraints which are non linear

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Linear Programming

Adopting the notation of matrix algebra a problem of linear programming(L.P.) in the canonical form can be written in the following way:

8>>>:max cTx

s.t.Ax bx 0

where z = cTx is the objective function to be maximized,

x = [x1, x2,.., xn] 2 Rn

is the vector of decision variables, b 2 Rm

is avector of known constants and c 2 Rn is a vector of coecients.

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Linear Programming

An L.P. problem can often appear in a dierent form than the canonicalone. However it is always possible to turn it in the canonical form. Amaximization problem can be turned into a minimization one observingthat:

min z = max (z)

and viceversamax z = min (z) .

Besides, all the contraints of the type Ax b can be rewritten as

Ax b; equality constraints Ax = b can be replaced with twoinequality contraints, imposing that at the same time Ax b andAx b.

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Linear Programming

Denition

An admissible solution to an L.P. problem is any vector x 2 Rn satisfyingthe system of constraints. The set of all the admissible solutions, which

usually named Sa is dened as Sa = fx 2 Rn

jx 0, Ax bg

Denition

An optimal solution to an L.P. problem is an admissible solution lettingthe objective function to assume the maximum nite value.

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Linear Programming - example

In this video we try to solve gracally the following optimization probelm:

Link:

http://www.engineeringvideos.net/optimization/optimization.php(University of Brescia) 8 / 42

E l S l
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Excel Solver

Excel Solver is a powerful tool for analyzing and solving varioustypes of linear programming problems which contain many variablesand constraints

It can analyze and solve linear programming (LP) problems, as wellas other kinds of mathematical programming problems (non-linear,

integer, mixed linear-integer,...).Because of its algorithmic nature, Excel solver may not alwaysconverge to the best solution

the result is inuenced by options such as the maximum time allowed,

the maximum number of iterations allowed, the size of the residualerror, etc.Excel solver provides default settings for all options, which areappropriate for most problems.


E l S l
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Excel Solver







E l S l
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Excel Solver







E l S l
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Excel Solver

Excel Solveris a powerful tool for analyzing and solving varioustypes of linear programming problems which contain many variables

and constraints






E l S l
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Excel Solver

Excel Solveris a powerful tool for analyzing and solving varioustypes of linear programming problems which contain many variables

and constraints






Excel Solver
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Excel Solver

Excel Solver (i.e. Risolutore) must be activated, if it is not alreadyactive


Excel Solver
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Excel Solver

Excel Solver (i.e. Risolutore) must be activated, if it is not alreadyactive

To activate the Analysis ToolPak, choose the Tools -> Data Analysiscommand (in Excel 2003) or click on Risolutore button (in Excel2007).


Excel Solver
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Excel Solver

The Solver interface to enter all the parameters of an L.P. problem:- the cell containing the objective function

- if the user wants to solve a max, min or equal to problem- enter the contraints

Figure: Solver interface


Linear Programming
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Linear ProgrammingCase: Acme Company

Among other products, Acme produces leg muscle vitamins (V) andjet-propelled tennis shoes (S)

The Acme Company products.


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Acme uses Excel solver to for the following LP problem:

maximize its total prot from assembling and packaging boxes of V

and S;decision variable is the number of units of V and S to be produced;

subject to the constraints on the use of its producing factors (plantsand workforce have a limited productivity).


Linear Programming


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Linear Programming


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Linear Programming
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Linear ProgrammingAcme Company: The LP mathematical formulation

Let us see the mathematical formulation of the objective function andconstraints. The decision variables appearing both in the objectivefunction and the constraints are

x1, which is the number of boxes of V,

x2, which is the number of boxes of SThe prot to be maximized is

Z = 2x1 + 3x2,

where2 is the unit prot of a box of V,3 is the unit prot of a box of S


Linear Programming


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Z = 2x1 + 3x2,



Linear Programming


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Z = 2x1 + 3x2,



Linear Programming
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g gAcme Company: The LP mathematical formulation




Z = 2x1 + 3x2,



Linear Programming
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Z = 2x1 + 3x2,



Linear Programming
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Z = 2x1 + 3x2,



Linear Programming
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The assembly time constraint is

9x1 + 15x2 108, 000

where

9 is the number of minutes required to assemble a box of V,

15 is the number of minutes required to assemble a box of S,108, 000 is the maximum number of minutes available in assembling

The packaging time constraint is

11x1 + 5x2 108, 000

where

11 is the number of minutes required to package a box of V,5 is the number of minutes required to package a box of S,108, 000 is the maximum number of minutes available in packaging


Linear Programming
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Acme Company: The LP mathematical formulation


9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear Programming
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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear Programming
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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear Programming
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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear Programming


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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear Programming


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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



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9x1 + 15x2 108, 000

where




11x1 + 5x2 108, 000

where



Linear ProgrammingA C Th LP h i l f l i


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The integer constraint is

x1, x2 2 Z+ [ f0g ,

and means that the numbers of boxes must be non negative andinteger


Linear ProgrammingA C Th LP E l S l f l i
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Acme Company: The LP Excel Solver formulation

The following picture shows the input data and the formulas required

to run the Excel Solver

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A B C D E F G H I

Acme Company

Assembly Packaging

Time (min) Time (min) Profit Changing Cells

leg muscle vitamins (V) 9 11 2 9'000jet-propelled tennis shoes (S) 15 5 3 1'800

Maximum time (min) 108'000 108'000

Actual time (min) 108'000 108'000

Target cell

23'400

Input data

Formulas

Cell Formula Copied to

C10 C6*$G6 + C7*$G7 D10

F12 E6*G6 + E7*G7Solver Parameters

F12

max

G6:G7

C10


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Changing cells: cells containing the decision variables. The changingcells are usually left blank or are assigned zero values

Target cell: cell containing the objective function in terms of thedecision variables

Constraints: cells containing the limits which are imposed upondecision variables


Linear ProgrammingAcme Company: The LP Excel Solver formulation
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Linear ProgrammingAcme Company: The LP Excel Solver formulation


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Linear ProgrammingAcme Company: Excel Solver at work


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Acme Company: Excel Solver at work

Activate the Excel Solver. Select Tools -> Solver to display the SolverParameters dialog box and follow what is written in the spreadsheet

Click the `Solve' button to start Solver

Select the `Keep Solver Solution' option on the `Solver Results' dialogbox in order to store the answer in the worksheet

For most problems, the default settings in the `Solver Options' boxare appropriate

The answer is 9, 000 boxes of V and 1, 800 boxes of S, giving a

maximum prot of 23, 400. The actual amount of time used in eachdepartment is 1, 800 hours, equalling the maximum time available


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p y








Main topics
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1 Linear Programming

2 StatisticsDescriptive StatisticsLinear RegressionProbabilityInference


StatisticsIntroduction and examples
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p

The primary role of statistics is to provide managers withmathematical tools that will help them to organize and analyze theinformation data in an eective and meaningful way

The easiest method of organizing data is to construct a frequencydistribution table using classes. Usually a class is an interval

One possible analysis on data is to test for the dierence amonggroups of data. The test can concentrate on the measures of position(mode, median, mean) and dispersion (range, standard deviation,

variance)


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variance)


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variance)


StatisticsWheelie Company: Excel FREQUENCY function and ChartWizard
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Consider the 100 numbers representing the length (in 100s

kilometers) travelled by bicycle tyres before one tyre of the couplefailed to meet minimum Wheelie Company standards

The Wheelie Company tyres.

We can organize the tyres survivorship into a frequency distribution.The rst task is to convert the raw data into a number of groups orclasses, and then count the number of values which fall into each class


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Consider the 100 numbers representing the length (in 100s

kilometers) travelled by bicycle tyres before one tyre of the couplefailed to meet minimum Wheelie Company standards

The Wheelie Company tyres.

We can organize the tyres survivorship into a frequency distribution.The rst task is to convert the raw data into a number of groups orclasses, and then count the number of values which fall into each class


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The FREQUENCY function needs two input ranges: the range of

data (RD) and the range of classes upper bounds (RCUB)

Given the minimum, m, and the maximum, M, of data, the classwidth, cw, is

cw =M m

N

,

where N is the desired number of classes. Each upper bound ubi,i = 1, ..., N, is calculated as

ubi = m + i cw

Select N vertically adjacent cells, write"=FREQUENCY(RD;RCUB)", then press "CTRL+ALT+ENTER"

Use ChartWizard to plot the frequency distribution


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cw =M m

N

,


ubi = m + i cw




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cw =M m

N

,


ubi = m + i cw




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cw =M m

N

,


ubi = m + i cw




Linear regressionIntroduction
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Among the models used to represent phenomena, the regression

models play a major roleRegression models allow the specication of a causal relationshipbetween a dependent variable and one or more independent variables.The former variable is to be explained in terms of the latter

A general regression (linear or non-linear) model is

Y = f(X1,..., Xp) + ,

where Y is the dependent variable, X1,..., Xp are the independentvariables, and is the error random variable

the error random variable is included to take into account

independent variables not included in the model on purpose, or simply

inadvertently

a non deterministic relationship, f, between X1, ...,Xp and Y

errors in the measurement of the variables

errors in the specication of f


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Y = f(X1,..., Xp) + ,




inadvertently





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Y = f(X1,..., Xp) + ,




inadvertently







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Y = f(X1,..., Xp) + ,




inadvertently





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Y = f(X1,..., Xp) + ,




inadvertently





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Y = f(X1,..., Xp) + ,




inadvertently





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Y = f(X1,..., Xp) + ,




inadvertently





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Y = f(X1,..., Xp) + ,




inadvertently





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f typically depends on parameters to be estimated based on a sample

of observations of Y and X1, ..., Xp

Y X1 ... Xpy1 x1,1 ... xp,1...

.... . .

...

yN x1,N ... xp,N

Estimating the parameters estimates in f yields the empirical model

Y = f(X1,..., Xp),

i.e. the formula for obtaining y1 from x1,1,..., xp,1, ..., yN fromx1,N,..., xp,N

Typically the parameters are estimated to make the dierencesy1 y1, ..., yN yN as small as possible


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Y X1 ... Xpy1 x1,1 ... xp,1...

.... . .

...

yN x1,N ... xp,N


Y = f(X1,..., Xp),




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Y X1 ... Xpy1 x1,1 ... xp,1...

.... . .

...

yN x1,N ... xp,N


Y = f(X1,..., Xp),




Linear regressionThe linear regression model
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The linear regression model is specied by the formula

Y = 0 +p

k=1

kXk +

Each parameter k tells how the dependent variable Y modies as Xkmodies by 1 unit, letting all other independent variables xed

Parameters 0,...,p are typically estimated with the least squaresmethod to yield the empirical model

Y = 0 +p

k=1

kXk


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Y = 0 +p

k=1

kXk +

Each parameter k tells how the dependent variableY

modies asXk

modies by 1 unit, letting all other independent variables xed


Y = 0 +p

k=1

kXk


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Y = 0 +p

k=1

kXk +

Each parameter k tells how the dependent variableY

modies asXkmodies by 1 unit, letting all other independent variables xed


Y = 0 +p

k=1

kXk


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Consider now the ith linear regression model

Yi = 0 +p

k=1

kXk,i + i, i = 1, ..., N,

i.e. the model referred to the ith observation of the variables,independently of the values yi, x1,i,..., xp,i, ei that will be observed.Classical assumptions about the linear regression model are

E(i) = 0Var(i) =

2

Cov(i, j), j = 1, ..., N, i 6= jX1,i,..., Xp,i are not random, i.e. they are xed at the observed valuesx1,i,..., xp,i




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Yi = 0 +p

k=1

kXk,i + i, i = 1, ..., N,


E(i) = 0Var(i) =

2



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Yi = 0 +p

k=1

kXk,i + i, i = 1, ..., N,


E(i) = 0Var(i) =

2



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Yi = 0 +p

k=1

kXk,i + i, i = 1, ..., N,


E(i) = 0Var(i) =

2



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Yi = 0 +p

k=1

kXk,i + i, i = 1, ..., N,


E(i) = 0Var(i) =

2


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To assess the linear regression model t, many indicators have beendeveloped. We see the R square indicator, R2

The more the model will adapt to data the lower the error variabilitywill be w.r.t. the total variability

The formula is

R2 = 1 RSS

TSS,

where RSS is the Residual Sum of Squares and TSS is the Total Sum

of Squares


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To assess the linear regression model t, many indicators have beendeveloped. We see the R square indicator, R2

The more the model will adapt to data the lower the error variabilitywill be w.r.t. the total variability

The formula is

R2 = 1 RSS

TSS,

where RSS is the Residual Sum of Squares and TSS is the Total Sum

of Squares


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Notice that R2 2 [0, 1] and the better the t is the closer to 1 R2 is

Notice also that R2 increases as p, the number of independentvariables, increases. Thus, an adjusted R square indicator has been

advancedR2adj = 1

RSSNp1TSSN1

.

The numbers N p 1 and N 1 are also called Residual Degreesof Freedom and Total Degrees of Freedom, respectively




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Notice that R2 2 [0, 1] and the better the t is the closer to 1 R2 is

Notice also that R2 increases as p, the number of independentvariables, increases. Thus, an adjusted R square indicator has been

advancedR2adj = 1

RSSNp1TSSN1

.

The numbers N p 1 and N 1 are also called Residual Degreesof Freedom and Total Degrees of Freedom, respectively


Linear regressionThe linear regression model at work: Excel Analysis ToolPak regression routine

Consider the data of the "Regression" sheet


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gIn Excel, choose Tools -> Data Analysis -> Regression:

input the ranges of the Y and Xs variables observations. In our case,the Y variable is the return of stock A and the X variable is the returnof Index SPPspecify the top-left cell where you would like the output to appear. Inour case, it is cell F7

check the Residuals option

Notice the values of cells R Square, Adjusted R Square, the columnSS including RSS and TSS, and the column Coecients including theestimates of parameters 0, intercept, and 1, X Variable 1The cell Standard Error is the estimate of the standard deviation ofthe model residuals; its formula is

2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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check the Residuals optionNotice the values of cells R Square, Adjusted R Square, the columnSS including RSS and TSS, and the column Coecients including theestimates of parameters 0, intercept, and 1, X Variable 1The cell Standard Error is the estimate of the standard deviation ofthe model residuals; its formula is

2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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In Excel, choose Tools -> Data Analysis -> Regression:



2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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2 =

rRSS

N 2=

svar(Residuals) N 1

N 2



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2 =

rRSS

N 2=

svar(Residuals) N 1

N 2


ProbabilityIntroduction

Decision-making means choosing between two or more alternatives.
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Good decision-making is based on evaluating which alternative has

the best chance of succeeding. When managers refer to the chance ofsomething occurring, they are using probability in the decision-makingprocessProbability is the chance (expressed with a real number p 2 [0, 1])

that something (an event) will happen

The ve Platonic solids and two



Decision-making means choosing between two or more alternatives.G
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Good decision-making is based on evaluating which alternative has

the best chance of succeeding. When managers refer to the chance ofsomething occurring, they are using probability in the decision-makingprocessProbability is the chance (expressed with a real number p 2 [0, 1])that something (an event) will happen

The ve Platonic solids and two

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Old unsatisfactory classical probability dened the probability of an


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Old, unsatisfactory classical probability dened the probability of an

event, given that each of the outcomes of an experiment are equallylikely, as

P(event) =number of outcomes realizing the event

total number of outcomes

Frequentist denition improves on classical weakness. It proposes todene probability of an event as a limit, in particular as the limit ofthe ratio between the number of time when the event happens overthe total number of observations:

limn!

nA

n= P(A)

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ProbabilityProbability distributions

A probability distribution is a way of recording the way probability
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distributes over each event linked to an experiment. We will brieydiscuss of four probability distributions. They are real probabilitydistributions because they are linked to events represented by sets ofreal numbers

the Binomial distribution, which is a discrete distribution

the Poisson distribution, which is a discrete distribution often used tocount the number of occurrences of some event in a given period oftimethe exponential distribution, which is a continuous distribution used tomeasure the length of time needed to perform some activity

the important continuous distribution known as the normal distributionThe cumulative probability distribution of real-dened events is theprobability of events dened by interval (, x], with x 2 R. Asx ! + we see that P((, x])! 1

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the important continuous distribution known as the normal distributionThe cumulative probability distribution of real-dened events is theprobability of events dened by interval (, x], with x 2 R. Asx ! + we see that P((, x])! 1

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the important continuous distribution known as thenormal distribution

The cumulative probability distribution of real-dened events is theprobability of events dened by interval (, x], with x 2 R. Asx ! + we see that P((, x])! 1

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A probability distribution is a way of recording the way probabilitydi ib h li k d i W ill b i
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the important continuous distribution known as thenormal distribution


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A probability distribution is a way of recording the way probabilitydi ib h li k d i W ill b i
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the Poisson distribution, which is a discrete distribution often used tocount the number of occurrences of some event in a given period oftimethe exponential distribution, which is a continuous distribution used tomeasure the length of time needed to perform some activitythe important continuous distribution known as the normal distribution




A probability distribution is a way of recording the way probabilitydi t ib t h t li k d t i t W ill b i
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the Poisson distribution, which is a discrete distribution often used tocount the number of occurrences of some event in a given period oftimethe exponential distribution, which is a continuous distribution used tomeasure the length of time needed to perform some activitythe important continuous distribution known as the normal distribution



ProbabilitySellEvryThing Company: BINOMDIST, Excel binomial distribution function

A salesman at SellEvryThing Company makes twenty calls per day torandomly selected homes The probability of the salesman making a
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randomly selected homes. The probability of the salesman making a

sale is 0.1, i.e.P(1) = 0.1,

where 1 indicates the event "a phone call ends with a sale"

The client answering the call of SellEvryThing

Company.


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Given that the probability of a successful outcome is p, the binomialdistribution indicates the probability of succeeding q times over ntrials (n q):

P(q) =n

q

pq(1 p)nq =n!

q!(n q)! pq(1 p)nq

Each trial is independent of the others

Each trial is also called Bernoulli trial

Each Bernoulli trial has two outcames: success, or fail


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P(q) =n

q

pq(1 p)nq =n!

q!(n q)! pq(1 p)nq





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P(q) =n

q

pq(1 p)nq =n!

q!(n q)! pq(1 p)nq





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P(q) =n

q

pq

(1 p)n

q

=n!

q!(n q)! pq

(1 p)n

q






Use Excel binomial distribution function BINOMDIST to nd theprobability of
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p y

no sales, i.e. P(0)four sales, i.e. P(4)more than four sales, i.e.

20

i=5

P(i) = P(5) + P(6) + ... + P(20)

= 1 P(0) P(1)

P(2) P(3) P(4) = 1 4

i=0

P(i) = 1 P((, 4])

four or more sales, i.e.20

i=4

P(i) = 1 3

i=0

P(i) = 1 P((, 3])



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p y


20

i=5

P(i) = P(5) + P(6) + ... + P(20)

= 1 P(0) P(1)

P(2) P(3) P(4) = 1 4

i=0

P(i) = 1 P((, 4])


i=4

P(i) = 1 3

i=0

P(i) = 1 P((, 3])



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y


20

i=5

P(i) = P(5) + P(6) + ... + P(20)

= 1 P(0) P(1)

P(2) P(3) P(4) = 1 4

i=0

P(i) = 1 P((, 4])


i=4

P(i) = 1 3

i=0

P(i) = 1 P((, 3])



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20

i=5

P(i) = P(5) + P(6) + ... + P(20)

= 1 P(0) P(1)

P(2) P(3) P(4) = 1 4

i=0

P(i) = 1 P((, 4])


i=4

P(i) = 1 3

i=0

P(i) = 1 P((, 3])



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20

i=5

P(i) = P(5) + P(6) + ... + P(20)

= 1 P(0) P(1)

P(2) P(3) P(4) = 1 4

i=0

P(i) = 1 P((, 4])


i=4

P(i) = 1 3

i=0

P(i) = 1 P((, 3])


InferenceIntroduction and sampling distribution
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Inferential statistics, usually abbreviated to inference, is a process bywhich conclusions about the features of a population arereached on the basis of examining only a part of it. Think to anopinion poll that is used to predict the voting pattern of a country'spopulation during an election

A quality-control manager will take a random sample of products andif it is found that the number of defective items is too high the entirebatch will be rejected. Label a defective item of the sample with 1,while put a 0 to each complying item. The sample mean of 1s and 0s

is the proportion of defective items


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Inferential statistics, usually abbreviated to inference, is a process bywhich conclusions about the features of a population arereached on the basis of examining only a part of it. Think to anopinion poll that is used to predict the voting pattern of a country'spopulation during an election

A quality-control manager will take a random sample of products andif it is found that the number of defective items is too high the entirebatch will be rejected. Label a defective item of the sample with 1,while put a 0 to each complying item. The sample mean of 1s and 0s

is the proportion of defective items




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When the mean is calculated from a sample, the observed value, X,depends on which sample was extracted (of the many possiblesamples that could be chosen).

Two samples from the same population are likely to have dierentsample means, therefore possibly leading to dierent conclusions

Managers need to understand how sample means are distributedthroughout the population, i.e. the sampling mean distribution.




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InferenceAstroReturns Company: sampling mean distribution

The investment manager of AstroReturns Company has been askedby a client to formulate a hypothesis about the average return on his
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portfolio investment of six stocks at the end of the next year

How stock markets are perceived nowadays more than ever.



The manager has the following data about the six stocks returns (in%) realized in the last year


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Stock A B C D E FReturn (%) 8 11 3 18 3 5

Let's illustrate the concept of sampling error. The investment

manager is shrewd and he will base his report on the best meanreturn of a sample of three stocks from the six available, i.e. stocksD, B, and A. The hypothesis he formulates to the client is:

next year's return on your portfolio investment of stocks A, B, C, D, E,

and F will be18 + 11 + 8

3= 12, 33%



The manager has the following data about the six stocks returns (in%) realized in the last year
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Stock A B C D E FReturn (%) 8 11 3 18 3 5

Let's illustrate the concept of sampling error. The investment

manager is shrewd and he will base his report on the best meanreturn of a sample of three stocks from the six available, i.e. stocksD, B, and A. The hypothesis he formulates to the client is:

next year's return on your portfolio investment of stocks A, B, C, D, E,

and F will be18 + 11 + 8

3= 12, 33%



However the client is not dumb, because he knows that this is onlyone of the possible 20 outcomes. See the following table
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Stock sample Mean return (%) Stock sample Mean return (%)

CEF 1.67 ABE 7.33ACE 2.67 ACD 7.67ACF 3.33 ABF 8.00

BCE 3.67 BCD 8.67BCF 4.33 DEF 8.67ABC 5.33 ADE 9.67AEF 5.33 ADF 10.33CDE 6.00 BDE 10.67

BEF 6.33 BDF 11.33CDF 6.67 ABD 12.33

where the 20 outcomes are calculated from the combinations (withoutrepetition) of 6 objects of class 3(University of Brescia) 41 / 42


To avoid "loosing his fees" for having been too optimistic, the


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investment manager should organize the sample means to have aclearer picture, i.e. he needs to draw the sampling mean distribution

Exercise: use Excel FRQUENCY function and ChartWizard torepresent the frequency distribution of the sample mean by groupingthe data into 5 intervals. Notice that the mean of the sample

mean distribution is equal to the mean of the population

Exercise: repeat the previous exercise on a table of means of samplesof four stocks

Exercise: repeat the previous exercise on a table of means of samples

of ve stocksNotice that the chart of the sampling mean distribution tends

to become more bell-shaped




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