Quantitative Technique I PGP 11-13

Dr. Rohit Joshi, IIM ShillongData, Models and DecisionsSession 3: DistributionsPGP 13-15Learning ObjectivesIn this session, you will learn:Distinguish between discrete random variables and continuous random variables.Know how to determine the mean and variance of a discrete distribution and continuous distributions.Identify the type of statistical experiments that can be described by discrete distribution and continuous distributions, and know how to work such problems.

Random variablesRandom VariablesDiscrete Random VariableContinuousRandom VariableWhat is a distribution?Describes the shape of a batch of numbers

Probability of happening of each outcome

Why Distribution?can serve as a basis for standardized comparison of empirical distributionscan help us estimate confidence intervals for inferential statisticsform a basis for more advanced statistical methodsfit between observed distributions and certain theoretical distributions is an assumption of many statistical proceduresDistributionsDiscrete distributionsBinomial DistributionHyper geometric DistributionPoisson Distribution

Continuous DistributionNormal DistributionUniform DistributionBeta Distribution (t, F, Chi square)Exponential DistributionImportant discrete probability distribution: The binomial7The Binomial Distribution: PropertiesA fixed number of observations, nex. 15 tosses of a coin; ten light bulbs taken from a warehouseTwo mutually exclusive and collectively exhaustive categoriesex. head or tail in each toss of a coin; defective or not defective light bulb; having a boy or girlGenerally called success and failureProbability of success is p, probability of failure is 1 pConstant probability for each observationThe outcome of one observation does not affect the outcome of the otherTwo sampling methodsInfinite population without replacementFinite population with replacement8Binomial distribution Take the example of 5 coin tosses. Whats the probability that you flip exactly 3 heads in 5 coin tosses? 9Binomial distribution, generally

1-p = probability of failurep = probability of successX = # successes out of n trialsn = number of trialsNote the general pattern emerging if you have only two possible outcomes (call them 1/0 or yes/no or success/failure) in n independent trials, then the probability of exactly X successes= 10Binomial distribution: exampleIf I toss a coin 20 times, whats the probability of getting exactly 10 heads?

11Binomial distribution: exampleIf I toss a coin 20 times, whats the probability of getting of getting 2 or fewer heads?

12**All probability distributions are characterized by an expected value and a variance:If X follows a binomial distribution with parameters n and p: X ~ Bin (n, p) Mean

Variance and Standard Deviation

Wheren = sample sizep = probability of success(1 p) = probability of failure13ApplicationsA manufacturing plant labels items as either defective or acceptableA firm bidding for contracts will either get a contract or notA marketing research firm receives survey responses of yes I will buy or no I will notNew job applicants either accept the offer or reject itYour team either wins or loses the football game at the company picnicThe Hypergeometric DistributionThe binomial distribution is applicable when selecting from a finite population with replacement or from an infinite population without replacement.

The hypergeometric distribution is applicable when selecting from a finite population without replacement.The Hypergeometric Distribution

WhereN = population sizeA = number of successes in the population N A = number of failures in the populationn = sample sizeX = number of successes in the sample n X = number of failures in the sampleThe Hypergeometric DistributionExampleDifferent computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded?So, N = 10, n = 3, A = 4, X = 2

The probability that 2 of the 3 selected computers have illegal software loaded is .30, or 30%.

The Hypergeometric DistributionCharacteristicsThe mean of the hypergeometric distribution is:

The standard deviation is:

Where is called the Finite Population Correction Factor

from sampling without replacement from a finite populationThe Poisson Distribution DefinitionsAn area of opportunity is a continuous unit or interval of time, volume, or such area in which more than one occurrence of an event can occur.

ex. The number of scratches in a cars paintex. The number of mosquito bites on a personex. The number of computer crashes in a day The Poisson Distribution PropertiesApply the Poisson Distribution when:You wish to count the number of times an event occurs in a given area of opportunityThe probability that an event occurs in one area of opportunity is the same for all areas of opportunity The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunityThe probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smallerThe average number of events per unit is (lambda)

The Poisson Distribution Formula

where:X = the probability of X events in an area of opportunity = expected number of eventse = mathematical constant approximated by 2.71828An exampleSuppose that, on average, 5 cars enter a parking lot per minute. What is the probability that in a given minute, 7 cars will enter?

So, there is a 10.4% chance 7 cars will enter the parking in a given minute.

Mean = Variance =

Continuous Probability DistributionContinuous DistributionA continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values)thickness of an itemtime required to complete a tasktemperature of a solutionheight

These can potentially take on any value, depending only on the ability to measure precisely and accurately.The Normal Distribution Properties Bell Shaped Symmetrical and asymptotic Mean, Median and Mode are equal Location is characterized by the mean, Spread is characterized by the standard deviation, Area to right and left of mean is 1/2.The random variable has an infinite theoretical range: - to + Mean = Median = Modef(X)The Normal Distribution Density FunctionThe formula for the normal probability density function is

Wheree = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159 = the population mean = the population standard deviationX = any value of the continuous variableSigma understanding of a NC

q = 99.7 %What is a Sigma level?A metric that indicates how well a process is performing.Higher is betterMeasures the capability of the process to perform defect-free workAlso known as z, it is based on standard deviation for continuous dataFinding ProbabilitiesProbability is the area under the curve!cdXf(X)

Many Normal Distribution

By varying the parameters and , we obtain different normal distributionsThere are an infinite number of normal distributions-3-2-10123Table Lookup of aStandard Normal Probability

Z0.00 0.01 0.02

0.000.00000.00400.00800.100.03980.04380.04780.200.07930.08320.0871

1.000.34130.34380.34611.100.36430.36650.36861.200.38490.38690.388812The Cumulative Standardized Normal DistributionZ.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.5478.020.1.5478Cumulative Standardized Normal Distribution Table (Portion) ProbabilitiesShaded Area ExaggeratedOnly One Table is Needed

Z = 0.12

Standardizing Example

Normal DistributionStandardized Normal DistributionShaded Area Exaggerated

Example:

Normal DistributionStandardized Normal DistributionShaded Area Exaggerated

Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.5832.020.1.5478Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated

Z = 0.21Example:

(continued)

Z.00.01-03.3821.3783.3745.4207.4168-0.1.4602.4562.45220.0.5000.4960.4920.4168.02-02.4129Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated

Z = -0.21Example:

(continued)

Example:

Normal DistributionStandardized Normal DistributionShaded Area Exaggerated

Example:

(continued)Z.00.010.0.5000.5040.5080.5398.54380.2.5793.5832.58710.3.6179.6217.6255.6179.020.1.5478Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated

Z = 0.30

.6217Finding Z Values for Known ProbabilitiesZ.000.20.0.5000.5040.50800.1.5398.5438.54780.2.5793.5832.5871.6179.6255.010.3Cumulative Standardized Normal Distribution Table (Portion)What is Z Given Probability = 0.6217 ?Shaded Area Exaggerated.6217

Recovering X Values for Known Probabilities

Normal DistributionStandardized Normal Distribution

An ExampleWe have a training program designed to upgrade the supervisory skills of production line supervisors. Because the program is self administered, supervisors require different no. of hours to complete the program. A study of past participation indicates that the mean length of time spent on the program is 500 hours and that this normally distributed random variable has a standard deviation of 100 hrs.Solve : (individual exercise)What is the probability that a participant selected at random will require more than 500 hrs to complete the program?Between 500 and 650 hrs to complete the training program?More than 700 hrs.Less than 580.Between 450 to 650.

Lowest Stock decision at post officeThe manager of a small postal substation is trying to quantify the variation in the weekly demand for mailing envelops. She has decided to assume that this demand is normally distributed. She knows that on an average 100 envelops are purchased weekly and that 90 percent of the time, weekly demand is below 115. The manager wants to stock enough mailing envelops each week so that the percentage of running out of envelops is no higher than 5 percent. Can you suggest her the lowest such stock level?Prediction of number of spectators in a matchMr. John, the McDonald stand manager for the One day Series at Sri Lanka's cricket stadium, just had two cancellation on his crew. This means that if more than 72,000 people come to watch todays cricket match, the line for hot-dogs will constitute a disgrace to Mr. John and will harm business at the future games. Mr. John knows from his experience that number of spectators who come to the game is normally distributed with mean 67,000 and standard deviation 4,000 people. Mr. John has an option to hire two temporary employees to ensure the business wont be harmed in the future at an additional cost of $200. If he believes the future harm to business of having more than 72,000 fans at the match would be $ 5000, what would you suggest him to go for?

Inspection ShopOn the basis of past experience, automobile inspectors in Maruti Udyog Limited in Gurgaon, have noticed that 5 percent of the cars coming in for their annual inspection fail to pass. Find the probability that between 7 and 18 of the next 200 cars to enter the Inspection shop will fail in the inspection.

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