Presentation on theme: "Chapter 6 Continuous Probability Distributions"— Presentation transcript:
1Chapter 6 Continuous Probability Distributions Uniform Probability DistributionNormal Probability DistributionExponential Probability Distributionxf (x)Exponentialf (x)xUniformxf (x)Normal
2Continuous 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 particular value.Instead, we talk about the probability of the random variable assuming a value within a given interval.
3Continuous Probability Distributions The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.x1x2Exponentialxf (x)f (x)xUniformx1x2xf (x)Normalx1x2
4Normal Probability Distribution The normal probability distribution is the most important distribution for describing a continuous random variable.It is widely used in statistical inference.
5Normal Probability Distribution It has been used in a wide variety of applications:Heightsof peopleScientificmeasurements
6Normal Probability Distribution It has been used in a wide variety of applications:TestscoresAmountsof rainfall
7Normal Probability Distribution CharacteristicsThe distribution is symmetric; its skewnessmeasure is zero.x
8Normal Probability Distribution CharacteristicsThe entire family of normal probabilitydistributions is defined by its mean m and itsstandard deviation s .Standard Deviation sxMean m
9Normal Probability Distribution CharacteristicsThe highest point on the normal curve is at themean, which is also the median and mode.x
10Normal Probability Distribution CharacteristicsThe mean can be any numerical value: negative,zero, or positive.x-1020
11Normal Probability Distribution CharacteristicsThe standard deviation determines the width of thecurve: larger values result in wider, flatter curves.s = 15s = 25x
12Normal Probability Distribution CharacteristicsProbabilities for the normal random variable aregiven by areas under the curve. The total areaunder the curve is 1 (.5 to the left of the mean and.5 to the right)..5.5x
13Normal Probability Distribution Characteristicsof values of a normal random variableare within of its mean.68.26%+/- 1 standard deviationof values of a normal random variableare within of its mean.95.44%+/- 2 standard deviationsof values of a normal random variableare within of its mean.99.72%+/- 3 standard deviations
14Normal Probability Distribution Characteristics99.72%95.44%68.26%xmm – 3sm – 1sm + 1sm + 3sm – 2sm + 2s
15Standard Normal Probability Distribution A random variable having a normal distributionwith a mean of 0 and a standard deviation of 1 issaid to have a standard normal probabilitydistribution.
16Standard Normal Probability Distribution The letter z is used to designate the standardnormal random variable.s = 1z
17Standard Normal Probability Distribution Converting to the Standard Normal DistributionWe can think of z as a measure of the number ofstandard deviations x is from .
18Standard Normal Probability Distribution Example: Pep ZonePep Zone sells auto parts and supplies includinga popular multi-grade motor oil. When thestock of this oil drops to 20 gallons, areplenishment order is placed.PepZone5w-20Motor Oil
19Standard Normal Probability Distribution Example: Pep ZoneThe store manager is concerned that sales are beinglost due to stockouts while waiting for an order.It has been determined that demand duringreplenishment lead-time is normallydistributed with a mean of 15 gallons anda standard deviation of 6 gallons.The manager would like to know theprobability of a stockout, P(x > 20).PepZone5w-20Motor Oil
20Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Stockout ProbabilityStep 1: Convert x to the standard normal distribution.z = (x - )/= ( )/6= .83Step 2: Find the area under the standard normalcurve to the left of z = .83.see next slide
21Standard Normal Probability Distribution PepZone5w-20Motor OilProbability Table forthe Standard Normal DistributionP(0 < z < .83)
22Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Stockout ProbabilityStep 3: Compute the area under the standard normalcurve to the right of z = .83.P(z > .83) = .5 – P(z < .83)==Probabilityof a stockoutP(x > 20)
23Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Stockout ProbabilityArea ==Area = .2967z.83
24Standard Normal Probability Distribution PepZone5w-20Motor OilStandard Normal Probability DistributionIf the manager of Pep Zone wants the probability of a stockout to be no more than .04, what should the reorder point be? Assuming that it has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.
25Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Reorder PointArea = .9600Area = .0400zz.04
26Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Reorder PointStep 1: Find the z-value that cuts off an area of .04in the right tail of the standard normaldistribution.We look up the complement of the tail area ( = .96)
27Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Reorder PointStep 2: Convert z.04 to the corresponding value of x.x = + z.05 = (6)= or 26A reorder point of 26 gallons will place the probabilityof a stockout during leadtime at (slightly less than) .04.
28Standard Normal Probability Distribution PepZone5w-20Motor OilSolving for the Reorder PointBy raising the reorder point from 20 gallons to26 gallons on hand, the probability of a stockoutdecreases from about .20 to .04.This is a significant decrease in the chance that PepZone will be out of stock and unable to meet acustomer’s desire to make a purchase.