probability distributions: part 2 bsad 30 dave novak source: anderson et al., 2013 quantitative...

Probability distributions: part 2

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

Overview

Continuous Probability DistributionsUniform Probability Distribution Normal Probability DistributionExponential Probability Distribution

Link to examples of types of continuous distributions

• http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm

2

http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm



Overview

We will briefly look at three “common” continuous probability examplesUniformNormalExponential

In statistical applications, it is not unusual to find instances of random variables that follow a continuous uniform, Normal, or Exponential probability distribution

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Overview

Uniform Normal Exponential

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f (x)f (x)

x x

Uniform

x

f (x)Normal

xx

f (x)f (x) Exponential

Probability distributions

Probability distributions are typically defined in terms of the probability density function (pdf) pdf for continuous function gives us the

probability that a value drawn from a particular distribution (x) is between two values

pdf for discrete function gives us the probability that x takes on a single value

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Probability distributions

In both the discrete and continuous case, the cumulative distribution function (cdf) gives us the probability that x is less than or equal to a particular value

pdf and cdf provide a different visual representation of the same variable, x

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Discrete probability distributions Example of discrete uniform pdf (6-sided

die)

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Discrete probability distributions Example of discrete uniform cdf (6-sided

die)

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Continuous probability distributions Example of normal pdf

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Continuous probability distributions Example of normal cdf

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Continuous probability distributions A continuous random variable can assume

any value in an interval on the real line or in a collection of intervals

It is not possible to talk about the probability of the random variable assuming a specific value

Instead, we talk about the probability of the random variable assuming a value within a given interval or range

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Continuous probability distributions Examples of continuous random variables

include the following:The number of ounces of soup contained in

a can labeled “8 oz.”The flight time of an airplane traveling from

Chicago to New YorkThe drilling depth required to reach oil in an

offshore drilling operation

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Continuous probability distributions The probability of the random variable

assuming a value within a given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2

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f (x)f (x)

x x

Uniform

x1 x1 x2 x2

x

f (x)Normal

x1 x1 x2 x2 x1 x1 x2 x2

Exponential

xx

f (x)f (x)

x2 x2

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Continuous Uniform probability distributions A random variable is uniformly distributed

whenever the probability that the variable will assume a value in any interval of equal length is the same for each intervalThe uniform probability density function is

where: a = smallest value the variable can assume

b = largest value the variable can assume

f (x) = 1/(b – a) for a < x < b = 0 elsewhere

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Flight time example

Let x denote the flight time of an airplane traveling from Chicago to New York. Assume that the minimum flight time is 2 hours and that the maximum flight time is 2 hours 20 minutes

Assume that sufficient actual flight data are available to conclude that the probability of a flight time between 120 and 121 minutes is the same as the probability of a flight time within any other 1-minute interval up to and including 140 minutes Probability of flight arriving 2 hours and 2 minutes after

take off is the same as probability of flight arriving 2 hours and 10 minutes after take off

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Flight time example

Uniform PDF

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f(x) = 1/20 for 120 < x < 140

= 0 elsewhere

where:

x = flight time in minutes

Flight time example

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f(x)f(x)

x x120120 130130 140140

1/201/20

Flight Time (mins.)Flight Time (mins.)

Flight time example

f(x)f(x)

x x120120 130130 140140

1/201/20


P(135 < x < 140) = 1/20(5) = .25P(135 < x < 140) = 1/20(5) = .25

What is the probability that a flight will take

between 135 and 140 minutes?

135135

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Flight time example

f(x)f(x)

x x120120 130130 140140

1/201/20


P(121 < x < 128) = 1/20(7) = .35P(121 < x < 128) = 1/20(7) = .35

What is the probability that a flight will take

between 121 and 128 minutes?

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Normal probability distributions The normal probability distribution is the

most important distribution for describing a continuous random variable

It is widely used in statistical inference as the assumption of normality underlies many standard statistical tests

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Normal probability distributions What does this mean in practice?

Most statistical tests employ the assumption of normality

Deviations from normally distributed data will likely render those tests inaccurate

Tests that rely on the assumption of normality are called PARAMETRIC tests

Parametric tests tend to be very powerful and accurate in testing variability in data

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Normal probability distributions What does this mean in practice?

You CANNOT use statistical tests that assume a normal distribution if the data you are analyzing do not follow a normal distribution (at least approximately)

You can TEST this assumptionIf data can not assumed to be normally

distributed, you will likely need to use NONPARAMETRIC tests that make no distributional assumptions

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Parametric vs nonparametric Describe two broad classifications of

statistical proceduresA very well known definition of

nonparametric begins “A precise and universally acceptable definition of the term ‘nonparametric’ is presently not available” (Handbook of Nonparametric Statistics, 1962, p. 2)

Thanks! That’s helpful…

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Parametric vs nonparametric In general, nonparametric procedures do

NOT rely on the shape of the probability distribution from which they were drawn

Parametric procedures do rely on assumptions about the shape of the probability distribution It is assumed to be a normal distributionAll parameter estimates (mean, standard

deviation) assume the data come from an underlying normally distributed population

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Parametric vs nonparametricAnalysis Parametric Nonparametric

1) Compare means between two distinct/independent groups Two-sample t-test Wilcoxon rank-sum test

2) Compare two quantitative measurements taken from the same individual Paired t-test Wilcoxon signed-rank test

3) Compare means between three or more distinct/independent groups

Analysis of variance (ANOVA) Kruskal-Wallis test

4) Estimate the degree of association between two quantitative variables Pearson coefficient of correlation Spearman’s rank correlation

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Source: Hoskin (not dated) “Parametric and Nonparametric: Demystifying the Terms”

Normal probability distributions Why should you care?

You want to know which set of tests (parametric –vs- nonparametric) are appropriate for the data you have

Use of an inappropriate statistical tests yields inaccurate or completely meaningless results

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Normal probability distributions Why should you care?

It’s not a matter of being “a little wrong” – you either use an appropriate statistical test correctly and have something meaningful to say about the data OR you use an inappropriate statistical test (or use it incorrectly), and have nothing accurate to say about the data at all!

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Normal probability distributions The normal distribution is used in a wide

range of “real world” applicationsHeight of peopleTest scoresAmount of rainfallScientific tests

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Normal probability distributions The normal PDF

2 2( ) / 21( )

2xf x e

= mean

= standard deviation

= 3.14159

e = 2.71828

where:

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Normal probability distributions Characteristics of normal PDF

The distribution is symmetric, and is bell-shaped

x

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Family of normal distributions defined by mean, µ, and standard deviation, s

Highest point is at the mean, which is also the median and mode

xMean m31


Mean can be any numerical value including negative, positive, or zero

-10 0 20x

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Standard deviation determines the width of the curve: larger s results in wider, flatter curves

s = 15

s = 25

x33


Approximately 68% of all values or a normally distributed RV are within (+/-) 1 sof the mean

Approximately 95.4% of all values or a normally distributed RV are within (+/-) 2 sof the mean

Approximately 99.7% of all values or a normally distributed RV are within (+/-) 3 sof the mean

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xm – 3s m – 1s

m – 2sm + 1s

m + 2sm + 3s

m

68.26%

95.44%

99.72%

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Probabilities for the normal random variable are given by areas under the curve

The total area under the curve is 1 (.5 to the left of the mean and .5 to the right)

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.5 .5

x

Normal probability distributions Percentile ranking

If a student scores 1 standard deviation above the mean on a test, then the student performed better than 84% of the class (0.5 + 0.34 = 0.84)

If a student scores 2 standard deviations above the mean on a test, then the student performed better than 98% of the class (0.5 + 0.477 = 0.977)

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Normal probability distributions An RV with a normal distribution with mean,

µ, = 0, and standard deviation, s, = 1 follows a standard normal distribution

The letter z is used to refer to a variable that follows the standard normal distribution

We can think of z as a measure of the number of

standard deviations a given variable, x, is from the mean,

zx

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Standard normal distribution No naturally measured variable has this

distribution, so why do we care about it?ALL other normal distributions are equivalent

to this distribution when the unit of measurement is changed to measure standard deviations from the mean

It’s important because ALL normal distributions can be “converted” to standard normal, and then we can use the standard normal table to find needed information

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Auto parts store example

Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed

The store manager is concerned that sales are being lost due to stockouts (running out of a product) while waiting for an order. It has been determined that customer demand during replenishment lead-time (the time it takes between when an order is placed and the order arrives at Pep Zone) is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.

The manager would like to know the probability of a stockout, P(x > 20)40

Auto parts store example stockout

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Auto parts example stockout Use the probability table for SND

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0 .83

Area = 0.5Area = 0.2967

So, 1 – (0.5+0.2967) = 1 - 0.7967 = .2033

z

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Auto parts store examplereorder pointIf the manager wants the probability of a stockout to be no more than 0.05 (5%), what is the appropriate reorder point?

The manager wants to minimize the risk of stocking out – which is currently 20%

If the manager sets the stockout probability threshold at 5%, what is the new reorder point? The existing reorder point is 20 gallons

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Auto parts store example reorder point

0

Area = .4500

Area = .05

z

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Area = .5

Auto parts example reorder point

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Auto parts store examplereorder point

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Auto parts store examplereorder point By increasing the reorder point from 20

gallons to 25 gallons, we can the probability of a stockout can be decreased from about .20 to .05 (20% to less than 5%)

This is a significant decrease in the chance that the store will be out of stock and unable to meet customer demand

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Auto parts store examplereorder point An obvious related question would be, what

have stockouts cost the store to date?How many sales $ has the store lost due to

stockouts?How many customers has the store lost due

to stockouts? Not just lost sales because the product the customer wants to purchase is not in stock, but how many of those customers never come back at all?

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Exponential probability distributions The exponential probability distribution is

also an important distribution for describing a continuous random variable

It is useful in describing the time it takes to complete at task:Time between arrivals at a check outTime between arrivals at a toll boothTime required to complete a questionnaireDistance between potholes in a roadway

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Similarity to Poisson distribution The Poisson distribution provides an

appropriate description of the number of occurrences per intervalDiscrete and can be counted

The exponential distribution provides an appropriate description of the length of the interval (time, distance, etc.) between occurrencesContinuous and needs to be measured

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Exponential probability distributions Exponential density function

where: = mean

e = 2.71828

f x e x( ) / 1

for x > 0, > 0

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Exponential probability distributions Cumulative density function

P x x e x( ) / 0 1 o

where:

x0 = some specific value of x

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Fueling example

The time between arrivals of cars at Al’s full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes

Al would like to know the probability that the time between any two successive arrivals will be 2 minutes or less

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Fueling example

xx

f(x)f(x)

.1.1

.3.3

.4.4

.2.2

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10Time Between Successive Arrivals (mins.)

P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866

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Summary

Examples of continuous probability distributionsUniformNormalExponential

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probability distributions: part 2 bsad 30 dave novak source: anderson et al., 2013 quantitative...

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