# Probability distributions: part 2

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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 Distributions
Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution Link to examples of types of continuous distributions

Overview We will briefly look at three “common” continuous probability examples Uniform Normal Exponential In statistical applications, it is not unusual to find instances of random variables that follow a continuous uniform, Normal, or Exponential probability distribution

Overview Uniform Normal Exponential x f (x) f (x) f (x) x x

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

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

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

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

Continuous probability distributions
Example of normal pdf

Continuous probability distributions
Example of normal cdf

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

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 York The drilling depth required to reach oil in an offshore drilling operation

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 f (x) x Uniform x1 x2 x f (x) Normal x1 x2 x1 x2 Exponential x f (x) 13

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 interval The uniform probability density function is f (x) = 1/(b – a) for a < x < b = elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume

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

Flight time example Uniform PDF f(x) = 1/20 for 120 < x < 140
= elsewhere where: x = flight time in minutes

Flight time example f(x) 1/20 x 120 130 140 Flight Time (mins.)

Flight time example What is the probability that a flight will take
between 135 and 140 minutes? f(x) P(135 < x < 140) = 1/20(5) = .25 1/20 x 120 130 135 140 Flight Time (mins.)

Flight time example What is the probability that a flight will take
between 121 and 128 minutes? f(x) P(121 < x < 128) = 1/20(7) = .35 1/20 x 120 130 140 Flight Time (mins.)

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

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

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 assumption If data can not assumed to be normally distributed, you will likely need to use NONPARAMETRIC tests that make no distributional assumptions

Parametric vs nonparametric
Describe two broad classifications of statistical procedures A 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…

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 distribution All parameter estimates (mean, standard deviation) assume the data come from an underlying normally distributed population

Parametric vs nonparametric
Analysis 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 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

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!

Normal probability distributions
The normal distribution is used in a wide range of “real world” applications Height of people Test scores Amount of rainfall Scientific tests

Normal probability distributions
The normal PDF where:  = mean  = standard deviation  = e =

Normal probability distributions
Characteristics of normal PDF The distribution is symmetric, and is bell-shaped x

Normal probability distributions
Characteristics of normal PDF 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 x Mean m

Normal probability distributions
Characteristics of normal PDF Mean can be any numerical value including negative, positive, or zero x -10 20

Normal probability distributions
Characteristics of normal PDF Standard deviation determines the width of the curve: larger s results in wider, flatter curves s = 15 s = 25 x

Normal probability distributions
Characteristics of normal PDF Approximately 68% of all values or a normally distributed RV are within (+/-) 1 s of the mean Approximately 95.4% of all values or a normally distributed RV are within (+/-) 2 s of the mean Approximately 99.7% of all values or a normally distributed RV are within (+/-) 3 s of the mean

Normal probability distributions
Characteristics of normal PDF 99.72% 95.44% 68.26% x m m – 3s m – 1s m + 1s m + 3s m – 2s m + 2s

Normal probability distributions
Characteristics of normal PDF 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) .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 ( = 0.84) If a student scores 2 standard deviations above the mean on a test, then the student performed better than 98% of the class ( = 0.977)

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, 

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

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)

Auto parts store example stockout

Auto parts example stockout
Use the probability table for SND

Auto parts store example stockout

Auto parts store example stockout
Area = So, 1 – ( ) = = Area = 0.5 z .83

Auto parts store example reorder point
If 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

Auto parts store example reorder point
Area = .4500 Area = .5 Area = .05 z

Auto parts example reorder point

Auto parts store example reorder point

Auto parts store example reorder 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

Auto parts store example reorder 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?

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 out Time between arrivals at a toll booth Time required to complete a questionnaire Distance between potholes in a roadway

Similarity to Poisson distribution
The Poisson distribution provides an appropriate description of the number of occurrences per interval Discrete and can be counted The exponential distribution provides an appropriate description of the length of the interval (time, distance, etc.) between occurrences Continuous and needs to be measured

Exponential probability distributions
Exponential density function for x > 0,  > 0 where:  = mean e =

Exponential probability distributions
Cumulative density function where: x0 = some specific value of x

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

Fueling example f(x) P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866
.1 .3 .4 .2 x Time Between Successive Arrivals (mins.)

Summary Examples of continuous probability distributions Uniform
Normal Exponential