Probability distributions: part 1 BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly.

Probability distributions: part 1

BSAD 30

Dave Novak

Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

Covered so far…

Chapter 1: IntroductionWhat is modelingTypes of modelsBasic problem formulationReview of basic linear (algebraic) problems

Chapter 2: Introduction to probabilityReview of probability concepts (complement,

union, intersection, conditional probability, joint probability table, independence, mutually exclusive)

2

Overview

Random Variables Discrete Probability Distributions

Uniform Probability Distribution Binomial Probability DistributionPoisson Probability Distribution

Link to examples of types of discrete distributions

• http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Discrete_distributions/Discrete_distributions.htm3

http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Discrete_distributions/Discrete_distributions.htm




Overview

We will briefly look at three “common” discrete probability examplesUniformBinomialPoisson

In business applications, we often find instances of random variables that follow a discrete uniform, binomial, or Poisson probability distribution

4

What is a random variable?

A random variable (RV) is a numerical description of the outcome of an experiment

Keep in mind that there is a difference between numeric variables and categorical variablesNumeric: temperature, speed, age,

monetized data, etc.Categorical: state of residence, gender,

blood type, etc.

5

What is a random variable?

Two types of random variables:Discrete

Continuous

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Random variables

7

Random variables

Question Random Variable x Type

Familysize

x = Number of dependents infamily reported on tax return

Discrete

Distance fromhome to store

x = Distance in miles fromhome to the store site

Continuous

Own dogor cat

x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

Discrete

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Example

Discrete random variable (RV) with a finite number of possible values

There is a readily identifiable upper bound to the number of TVs sold on any given day

In this case, no more than 4 TVs sold

Let x = number of TVs sold at the store in one day,

where x can take on 5 values (0, 1, 2, 3, 4)

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Example

Discrete random variable (RV) with an infinite number of possible values

There is no readily identifiable upper bound on the number of customers coming into the store on any given day

There cannot be an infinite # of customers, but we are not setting an upper bound (could be 75, 500, or 2,000)

Let x = number of customers arriving in one day,

where x can take on the values 0, 1, 2, . . .

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Discrete probability distributions The probability distribution for a random

variable describes how probabilities associated with each value are distributed (or allocated) over all possible values

We can describe a discrete probability distribution with a table, graph, or equationIn the TV sales example, we would want a

mathematical and/or visual representation of the probability of selling 0, 1, 2, 3, or 4 TVs on any given day

11

Discrete probability distributions The probability distribution is defined by a

probability function, denoted by f(x), which provides the probability for each value of the random variableThe function f(x) is a mathematical

representation of the probability distributionThe following conditions are required:

f(x) > 0

f(x) = 112

Discrete distribution: DiCarlo motors example Using historical data on car sales, a tabular

representation of sales is created

Number Units Sold of Days

0 54 1 117 2 72 3 42 4 12

5 3 300

x f(x) 0 .18 1 .39 2 .24 3 .14 4 .04 5 .01 1.00

.18 = 54/300

.04 = 12/300

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Discrete distribution: DiCarlo motors example Graphical representation

.10.10

.20.20

.30.30

.40.40

.50.50

0 1 2 3 4 50 1 2 3 4 5Values of Random Variable x (car sales)Values of Random Variable x (car sales)

Pro

babi

lity

Pro

babi

lity

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Discrete distribution: DiCarlo motors example The probability distribution provides the

following informationThere is a 0.18 probability that no cars will

be sold during a day f(0) = 18%The most probable sales volume is 1, with

f(1) = 0.39 f(1) = 39%There is a 0.05 probability of either four or

five cars being sold f(4) + f(5) = 5%

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Summary Up to this point, we have not discussed

the specific TYPE of discrete probability distribution (i.e. uniform, binomial, Poisson, etc.)

We have only discussed probability distributions in terms of being discrete as opposed to continuous

A review of basic statistical concepts is next

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Expected value and variance The expected value, or mean, of a random

variable is a measure of its central locationMean, median, and mode are measures of

central tendency because they identify a single value as “typical” or representative of all values in a probability distribution

E(x) = = x f(x)

17

Expected value and variance The variance, 2, summarizes the variability

in the values of a random variable The standard deviation, , is defined as the

positive square root of the variance

Var(x) = 2 = (x - )2f(x)

StdDev(x) = =

18

Expected value and variance Both the StdDev and variance provide a

measure of how much the values in the probability distribution differ from the mean

The higher the standard deviation, the more different the different observations are from one another and from the mean

When a probability distribution has a high standard deviation, the mean is not a good measure of central tendency

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Expected value and varianceScores = 1,4,3,4,2,7,18,3,7,2,4,3Mean = 5Median = 3.5Standard Deviation = 4.53

The standard deviation indicates that the average difference between each score and the mean is around 4.5 points. However, only one score (18) is 4.5 or more points different from the mean. The one extreme score (18) overly influences the mean. The median (3.5) is a better measure of central tendency in this case because extreme scores do not influence the median

20

Discrete distribution: DiCarlo motors example

Number Units Sold of Days

0 54 1 117 2 72 3 42 4 12

5 3 300

x f(x) 0 .18 1 .39 2 .24 3 .14 4 .04 5 .01 1.00

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DiCarlo motors example

Calculate expected value of discrete RV

expected number of cars sold in a day

x f(x) xf(x) 0 .18 .00 1 .39 .39 2 .24 .48 3 .14 .42 4 .04 .16 5 .01 .05

E(x) = 1.50

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0 x 0.18 = 0

1 x 0.39 = 0.39


Calculate variance and StdDev

012345

-1.5-0.5 0.5 1.5 2.5 3.5

2.25 0.25 0.25 2.25 6.2512.25

.18

.39

.24

.14

.04

.01

.4050

.0975

.0600

.3150

.2500

.1225

x - (x - )2 f(x) (x - )2f(x)

Variance of daily sales = s 2 = 1.2500

x

carssquared

Standard deviation of daily sales = s = = 1.118 cars

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Calculate variance and StdDev

Standard deviation of daily sales = s = = 1.118 cars

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Var(x) = 2 = (x - )2f(x) = 0.4050 + 0.0975 + 0.0600 + 0.3150 + 0.2500 + 0.1225

Var(x) = 2 = 1.25

Expected value and variance From a decision-making or analyst

perspective what are some of the practical implications of this discussion?If the data you are analyzing have a high

variance, making decisions based on the mean, or even stressing the importance of the average, is likely to be misleading

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Expected value and variance What should you do?

Generate a visual representation of the data!You need to better characterize the data to

see if they fit into any well-known families of probability distributions – this would be the first step in analysis• Knowing what the data “aren’t” is also useful

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Expected value and variance What should you do?

Knowing that data do not follow a particular distribution is important in terms of analysis

There are particular characteristics associated with different types of distributions that can guide you in your analysis

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Discrete Distributions we will examine 1) Uniform

2) Binomial or Bernoulli

3) Poisson

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Discrete uniform probability distribution The discrete uniform probability distribution

is the simplest example of a discrete probability distribution given by a formula

Example: getting a 1, 2, 3, 4, 5, or 6 when rolling single die – f(x) = 1/6

f(x) = 1/n

where:n = the number of values the random variable may assume

the values of the random variable are equally likely

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Binomial probability distribution Also known as Bernoulli distribution Has four properties:

1) Experiment consists of n, independent trials

2) Only TWO outcomes are possible for each trial (success/failure, good/bad, on/off, yes/no, etc.)

3) The probability of success stays the same for all trials

4) All trials are independent30

Binomial probability distribution We are interested in the number of

successes, or positive outcomes occurring in the n trialsx denotes the number of successes, or

positive outcomes occurring in the n trials

where: f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

( )!( ) (1 )

!( )!x n xn

f x p px n x

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Binomial probability distribution

( )!( ) (1 )

!( )!x n xn

f x p px n x

Probability of a particular sequence of trial outcomes with x successes in n trials

Number of experimental outcomes providing exactlyx successes in n trials

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Binomial probability distribution Assume the probability that any customer

who comes into a store and actually makes a purchase is 0.3 (30% chance of success)

What is the probability that 2 of the next 3 customers who enter the store make a purchase?

Identify: n, x, p

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Binomial probability distribution (decision tree)

1st Customer 1st Customer 2nd Customer2nd Customer 3rd Customer3rd Customer xx Prob.Prob.

Purchases (.3)Purchases (.3)

(.7)Does NotPurchase

(.7)Does NotPurchase

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22

00

22

22



DNP (.7)DNP (.7)

Does NotPurchase (.7)Does NotPurchase (.7)

Does NotPurchase (.7)Does NotPurchase (.7)

DNP (.7)DNP (.7)

DNP (.7)DNP (.7)

DNP (.7)DNP (.7)

P (.3)P (.3)

P (.3)P (.3)

P (.3)P (.3)

P (.3)P (.3) .027.027

.063.063

.063.063

.343.343

.063.063

11

11

.147.147

.147.147

.147.147

11

35

Binomial probability distribution If a six-sided die is rolled three times, what

is the probability that the number 5 comes up twice?Identify: n, x, p

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37


1st roll 1st roll 2nd roll2nd roll 3rd roll3rd roll xx Prob.Prob.

Success “5” (.17)Success “5” (.17)

(.83)Failure (1,2, 3, 4, 6)

(.83)Failure (1,2, 3, 4, 6)

33

22

00

22

22

Success (.17)Success (.17) F (.83)F (.83)

Failure (.83)Failure (.83)

S (.17)S (.17) .005.005

.572.572

.024.024

11

11

.117.11711

Success (.17)Success (.17)

Failure (.83)Failure (.83)

S (.17)S (.17)

S (.17)S (.17)

S (.17)S (.17)

F (.83)F (.83)

F (.83)F (.83)

F (.83)F (.83)38

.024.024

.024.024

.117.117

.117.117

Binomial probability distribution What’s the probability if I roll a die 10 times,

the number 5 comes up four times?Identify: n, x, p

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Binomial probability distribution Expected value

Variance

Standard deviation

E(x) = = np

Var(x) = 2 = np(1 - p)

(1 )np p 40

Binomial probability distribution In the clothing store example, calculate:

Expected value

Variance

Standard deviation

41

Poisson probability distribution A Poisson distributed random variable is

often useful in estimating the number of occurrences over a specified interval of time or space which can be counted in whole numbersVery useful in RISK analysis

It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ∞)

42

Poisson probability distribution How is an RV that follows a Poisson

distribution different from an RV that follows a binomial distribution?It is possible to count how many events have

occurred, but meaningless to ask how many events have NOT occurred

In the binomial situation, we know the probability of two mutually exclusive events (p, q) – in the Poisson situation, we have no q (it has only one parameter the average frequency an event occurs)43

Poisson probability distribution Examples

Number of customers arriving at a supermarket checkout between 5 PM and 6 PM

Number of text messages you receive over the course of a week

Number of car accidents over the course of a year

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Poisson probability distribution Two properties of Poisson distributions

1) The probability of occurrence is the same over any two time intervals of equal length

2) The occurrence or nonoccurrence in any time interval is independent of occurrence or nonoccurrence in any other time interval

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Poisson probability distribution

!)(

x

exf

x

where:

f(x) = probability of x occurrences in an interval

l = mean number of occurrences in an interval

e = 2.71828

46

For more info: https://en.wikipedia.org/wiki/E_(mathematical_constant)

Drive-up teller window exampleSuppose that we are interested in the number of cars arriving at the drive-up teller window of a bank during a 15-minute period on weekday mornings We assume that the probability of a car arriving is the

same for any two time periods of equal length (i.e. prob of a car arriving in the first minute is exactly the same as the prob of a car arriving in the last minute), and the arrival or non-arrival of a car in any time period is independent of the arrival or non-arrival in any other time period

An analysis of historical data shows that the average number of cars arriving during a 15-minute interval of time is 10, so the Poisson probability function with = 10 applies47

Drive-up teller window example

l = 10 arrivals / 15 minutes, x = 5

We want to know the probability that exactly 5 cars will arrive over the 15 minute time interval

Identify: x and

X = 5

=> we are given that there are 10 arrivals every 15 minutes, so the average # of arrivals over the time period is 10

48

Drive-up teller window example

5 1010 (2.71828)(5) .0378

5!f

l = 10 arrivals / 15 minutes, x = 5

So, there is a 3.78% chance that exactly 5 cars will arrive over the 15 minute time period

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Highway defect example

• Suppose that we are concerned with the occurrence of major defects in a section of highway one month after that section was resurfaced

• We assume that the probability of a defect is the same for any two highway intervals of equal length (i.e. the probability of a defect between mile markers 1 and 2 is the same as the probability of a defect between mile markers 4 and 5, etc.) and that the occurrence of a defect in any one mile interval is independent of the occurrence or nonoccurrence of a defect in any other interval

• Thus, the Poisson probability distribution applies

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Find the probability that no major defects occur in a specific 3-mile stretch of highway assuming that major defects occur at the average rate of two defects per mile

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Poisson probability distribution Expected value

Variance

Standard deviation

E(x) = µ = the rate or frequency of an event

Var(x) = 2 =

=

53


In the highway defect example, calculate:Expected value

Variance

Standard deviation

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Summary

Discussion of random variables DiscreteContinuous

Examples of discrete probability distributionsUniformBinomialPoisson

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Probability distributions: part 1 BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly.

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