Probability distributions: part 2 BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

Last class Random Variables Discrete Continuous Discrete Probability Distributions Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution 2

Overview Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution Link to examples of types of continuous distributions Model_Assist.htmhttp:// Model_Assist.htm 3

Overview We will briefly look at three “commonly” observed continuous probability examples Uniform Normal Exponential In real-world applications, it is fairly common to find instances of random variables that follow a continuous uniform, Normal, or Exponential probability distribution 4

Overview Uniform Normal Exponential 5 f (x) x x Uniform x Normal x x Exponential

Review A random variable (RV) is a numerical description of the outcome of an experiment If an RV can take on ANY value within a range, it is continuous (measured) Any value between 0 and 100 (i.e , , etc.) If an RV must take on a well defined value within a range, it is discrete (counted) Must be an even integer between 0 and 100 (i.e. 4, 28, 92, etc.) 6

Review Just like RVs, probability distributions are classified as discrete or continuous Probability distributions: graphical, tabular, and/or mathematical representations that show the relationship between the possible outcomes of a statistical experiment, and the probability that each of those outcomes will occur 7

Probability distributions Probability distributions are typically defined in terms of the probability density function (pdf) pdf for discrete RV gives the probability that a value drawn from a particular distribution (x) takes on a particular value pdf for continuous RV gives the probability that a value drawn from a particular distribution (x) is between two values 8

Probability distributions pdf for discrete RV P(x) is equal to x pdf for continuous RV P(x) lies between a (upper bound) and b (lower bound) of some function The probability that a continuous RV is exactly equal to a particular value is zero! Why? 9

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 10

Discrete probability distributions Example of discrete uniform pdf for 6-sided die – mathematical representation Set of possible values X = {1, 2, 3, 4, 5, 6} x X Probability of value x: P(x) 11 x P(x)1/6

Discrete probability distributions Example of discrete uniform pdf for 6-sided die – graphical representation 12

Discrete probability distributions Example of discrete uniform cdf for 6-sided die 13

Discrete probability distributions Example of discrete pdf for two 6-sided dice – mathematical representation 14 Source:

Discrete probability distributions Example of discrete pdf for two 6-sided dice – graphical representation 15 Source:

Discrete probability distributions Example of cdf for two 6-sided dice 16 Source:

Continuous probability distributions Example of normal pdf 17

Continuous probability distributions Example of normal cdf 18

Continuous probability distributions A continuous RV can assume any value in an interval It is not possible to talk about the probability of a continuous RV assuming a specific value Given the range 0 ≤ x ≤ 1, what is the probability of x = 0.254? 19

Continuous probability distributions Instead, we talk about the probability of the random variable assuming a value within a given interval or range Given the range 0 ≤ x ≤ 1, what is the probability that 0.5 ≤ x ≤ 0.75? 20

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 21

Continuous probability distributions The probability of the random variable assuming a value within a given interval from x 1 to x 2 is defined to be the area under the graph of the pdf between a and b 22 f (x) x x Uniform a aa a a aa a b bb b b bb b x Normal a aa a a aa a b bb b b bb b x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x b bb b b bb b 22 a

Continuous Uniform probability distributions A continuous RV is uniformly distributed when 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 where: a = smallest value the variable can assume b = largest value the variable can assume f (x) = 1/(b – a) for a < x < b f (x) = 1/(b – a) for a < x < b = 0 elsewhere = 0 elsewhere f (x) = 1/(b – a) for a < x < b f (x) = 1/(b – a) for a < x < b = 0 elsewhere = 0 elsewhere 23

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 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 24

Flight time example 25 where: x = flight time in minutes Uniform pdf (mathematical representation Uniform pdf (graphical representation We are subdividing this area into 20 time intervals of 1 minute each The probability that the flight arrives is the same for all 20 of those intervals

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

Flight time example f(x)f(x) f(x)f(x) x x /20 Flight Time (mins.) P(121 < x < 128) = 1/20(7) =.35 What is the probability that a flight will take between 121 and 128 minutes? 27

Normal probability distributions The normal probability distribution is the most important distribution for describing a continuous RV It is widely used in statistical inference as the assumption of normality underlies many standard statistical tests 28

Normal probability distributions What does the assumption of normality 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 29

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 the normality assumption If data are not assumed to be normally distributed, you will likely need to use NONPARAMETRIC tests that make no distributional assumptions 30

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… 31

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 32

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 33 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 34

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 that can accurately be said about the data at all! 35

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 36

Normal probability distributions The normal PDF  = mean  = standard deviation  = e = where: 37

Normal probability distributions Characteristics of normal PDF Symmetric and is bell-shaped 38

Normal probability distributions Characteristics of normal PDF Family of normal distributions defined by mean, µ, and standard deviation,  Highest point is at the mean, which is also the median and mode x Mean  39

Normal probability distributions Characteristics of normal PDF Mean can be any numerical value including negative, positive, or zero x 40

Normal probability distributions Characteristics of normal PDF Standard deviation determines the width of the curve: larger  results in wider, flatter curves 41

Normal probability distributions Characteristics of normal PDF Approximately 68% of all values or a normally distributed RV are within (+/-) 1  of the mean Approximately 95.4% of all values or a normally distributed RV are within (+/-) 2  of the mean Approximately 99.7% of all values or a normally distributed RV are within (+/-) 3  of the mean 42

Normal probability distributions Characteristics of normal PDF 43

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) 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) 45

Normal probability distributions An RV with a normal distribution with mean, µ, = 0, and standard deviation, , = 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,  46

Standard normal distribution No naturally measured variable has this distribution, so why do we care about it? ALL 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 47

Auto parts store example Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the on hand inventory of oil drops to 20 gallons, a replenishment order is placed. The manager is concerned that sales are being lost due to stockouts (running out of a product) while waiting for the replenishment order to be filled. It’s estimated that customer demand during replenishment lead-time (the time between when the order is placed and the order arrives) is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. What is the probability of a stockout, P(x > 20)? 48

Auto parts store example stockout 49

Auto parts store example stockout 50

Auto parts example stockout Use the probability table for SND 51

Auto parts store example stockout Area = 0.5 Area = So, 1 – ( ) = =. =.2033 z 52

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  so, what should our ideal reorder point be, if we want to reduce the probability of a stockout from 20% to 5%? 53

Auto parts store example reorder point 0 Area =.4500 Area =.05 z 54 Area =.5

Auto parts example reorder point 55

Auto parts store example reorder point 56

Auto parts store example reorder point 57

Auto parts store example reorder point By increasing the reorder point from 20 gallons to 25 gallons, 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 probability that the store will be out of stock and unable to meet customer demand 58

Auto parts store example reorder point An obvious related question would be, what have stockouts cost the store to date? How many direct sales $ has the store lost due to stockouts? How many indirect sales $ 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 would have also made other purchases or never come back at all? 59

Exponential probability distributions The exponential probability distribution is also an important distribution for describing a continuous RV It is useful in describing the time it takes to complete at task, how much time elapses before an event occurs, distance between events, etc.: 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 60

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 61

Exponential probability distributions Exponential density function where:  = mean e = for x > 0,  > 0 62

Exponential probability distributions Cumulative density function where: x 0 = some specific value of x 63

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 64

Fueling example 65

Fueling example x x f(x)f(x) f(x)f(x) Time Between Successive Arrivals (mins.) P(x < 2) = /3 = =

Summary Examples of continuous probability distributions Uniform Normal Exponential 67