Chapter 6: Continuous Probability Distributions GBS221, Class March 25, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College

Transcription

1 Chapter Objectives 1. Understand the difference between hw prbabilities are cmputed fr discrete and cntinuus randm variables. 2. Knw hw t cmpute prbability values fr a cntinuus unifrm prbability distributin and be able t cmpute the expected value and variance fr such a distributin. 3. Be able t cmpute prbabilities using a nrmal prbability distributin. Understand the rle f the standard nrmal distributin in this prcess. 4. Be able t use tables fr the standard nrmal prbability distributin t cmpute standard nrmal prbabilities and prbabilities fr any nrmal distributin. 5. Be able t use Excel's NORMSDIST and NORMDIST functins t cmpute prbability fr the standard nrmal distributin and any nrmal distributin. 6. Be able t use Excel's NORMSINV and NORMINV functin t find z and x values crrespnding t given cumulative prbabilities. 7. Be able t cmpute prbabilities using an expnential prbability distributin and understand Excel's EXPONDIST functin. 8. Understand the relatinship between the Pissn and expnential prbability distributin. 1. Cntinuus Prbability Distributins A cntinuus randm variable can assume any value in an interval n the real line r in a cllectin f intervals. It is nt pssible t talk abut the prbability f the randm variable assuming a particular value. Instead, we talk abut the prbability f the randm variable assuming a value within a given interval. The prbability f the randm variable assuming a value within sme given interval frm x 1 t x 2 is defined t be the area under the graph f the prbability density functin between x 1 and x 2. The ttal area under the graph f(x) is equal t Area as a Measure f Prbability The area under the graph f f(x) and prbability are the same. The prbability that x takes a value between sme lwer value x 1 and sme higher value x 2 can be fund by cmputing the area under the graph f f(x) ver the interval x 1 and x 2. Unifrm Prbability Distributin If a randm variable is restricted t be within sme interval and the prbability density functin is cnstant ver the interval, (a,b), the cntinuus randm variable is said t have a unifrm distributin between a and b. Its density functin is given by: The frmula fr the expected value and variance f x is:

2 3. & 4. Nrmal Prbability Distributin The nrmal distributin is perhaps the mst widely used distributin fr describing a cntinuus randm variable. The nrmal prbability density functin is: Characteristics f the Nrmal Prbability Distributin The shape f the nrmal curve is ften illustrated as a bell-shaped curve. Tw parameters, µ (mean) and σ (standard deviatin), determine the lcatin and shape f the distributin. The highest pint n the nrmal curve is at the mean, which is als the median and mde. The mean can be any numerical value: negative, zer, r psitive. The nrmal curve is symmetric (left and right halves are mirrr images). The standard deviatin determines the width f the curve: larger values result in wider, flatter curves. The ttal area under the curve is 1 (0.5 t the left f the mean and 0.5 t the right). Areas under the curve give prbabilities fr the nrmal randm variable. Unlike the unifrm distributin, the height f the nrmal distributin's curve varies and calculus is required t cmpute the areas that represent prbability.. Areas under the nrmal curve have been cmputed and are available in tables that can be used in cmputing prbabilities. The percentage f values in sme cmmnly used intervals are: 68.26% f values f a nrmal randm variable are within +/- 1 standard deviatin f its 95.44% f values f a nrmal randm variable are within +/- 2 standard deviatins f its 99.72% f values f a nrmal randm variable are within +/- 3 standard deviatins f its Standard Nrmal Prbability Distributin A randm variable that has a nrmal distributin with a mean f zer and a standard deviatin f ne is said t have a standard nrmal prbability distributin. The letter z is cmmnly used t designate this nrmal randm variable. We can think f z as a measure f the number f standard deviatins x is frm µ. The frmula used t cnvert any nrmal randm variable x, with mean µ and standard deviatin σ, t the standard nrmal distributin is:

3 5. Using NORMDIST and NORMSDIST Using Excel s NORMDIST Functin The NORMDIST functin cmputes the cumulative prbability fr any nrmal distributin. We d nt need t cnvert an x value t a z value. The NORMDIST functin requires fur arguments The x value fr which we want t cmpute the cumulative prbability The mean The standard deviatin A value f TRUE r FALSE (Nrmally we put TRUE because we want a cumulative prbability) T cmpute a prbability f a randm variable assuming a value less than the x value, we use the functin =NORMDIST. T cmputer a prbability f a randm variable assuming a value greater than the x value, we enter the frmula =1-NORMDIST Using Excel s NORMSDIST Functin The NORMSDIST functin is used t cmpute the cumulative prbability fr the standard nrmal distributin. Essentially, we are cnverting a nrmal distributin t the standard nrmal distributin by cmputing and using a z value rather than an x value. The NORMSDIST functin requires ne argument, which is the z value. T cmpute a prbability f a randm variable assuming a value less than the x value, we use the functin =NORMSDIST. T cmputer a prbability f a randm variable assuming a value greater than the x value, we enter the frmula =1-NORMSDIST. 6. Using NORMINV and NORMSINV Using Excel s NORMINV Functin The NORMINV functin is used t cmpute the x value fr a given cumulative prbability fr any nrmal distributin. The NORMINV functin requires three arguments, which are the cumulative prbability, the mean, and the standard deviatin. T cmpute the x value, since we re slving fr it, we enter the frmula =NORMINV int the cell. Using Excel s NORMSINV Functin The NORMSINV functin is used t cmpute the x value fr a given cumulative prbability fr the standard nrmal distributin. The NORMSINV functin requires three arguments, which are the cumulative prbability, the mean, and the standard deviatin. T cmpute the x value, since we re slving fr it, we enter the frmula =NORMSINV int the cell. 7. Expnential Prbability Distributin A cntinuus prbability distributin frequently used fr cmputing the prbability f the time t cmplete a task is the expnential distributin. If the average time t cmplete a task is dente by p, then the prbability density functin fr the amunt f time, x, t cmplete the task is given by: Frm this distributin, the prbability a task is cmpleted within a specified time, x 0, is:

4 Using Excel s EXPONDIST Functin Excel s EXPONDIST functin can be used t cmpute expnential prbabilities. The EXPONDIST functin has three arguments: the first is the value f x 0, the secnd is the value f 1/µ, and the third is TRUE r FALSE. We will always select true because we are seeking a cumulative prbability. 8. Relatinship Between the Pissn and Expnential Distributins If the Pissn distributin prvides an apprpriate descriptin f the number f ccurrences per interval, the expnential distributin prvides a descriptin f the length f the interval between ccurrences. As an example: If custmers arrive accrding t a Pissn distributin with a mean f λ custmers, then the interarrival times f custmers fllws an expnential distributin with µ = 1/λ Slving fr an Upper Tail Area n a Nrmal Distributin Pep Zne sells aut parts and supplies including a ppular multi-grade mtr il. When the stck f this il drps t 20 gallns, a replenishment rder is placed. The stre manager is cncerned that sales are being lst due t stckuts while waiting fr a replenishment rder. It has been determined that leadtime demand is nrmally distributed with a mean f 15 gallns and a standard deviatin f 6 gallns. What is the prbability f a stckut P(x > 20)? We are slving fr the (tail) area t the right f the 20-galln line in the graph belw. The Cumulative Prbabilities fr the Standard Nrmal Distributin table in the textbk des nt directly prvide the area fr the upper tail regin f the distributin. S, we will first determine the area under the curve t the left f the rerder pint. We begin by cnverting ur nrmal distributin (measured in gallns) t the standard nrmal distributin s that we can use the Standard Nrmal table f areas. Essentially, we must cmpute hw many standard deviatins lie between the mean demand value (15) and the rerder pint value x (20). z = (x - µ)/σ = (20 15)/6 =.83 The Cumulative Prbabilities fr the Standard Nrmal Distributin table shws an area f.7967 fr the regin t the left f the rerder pint where z =.83. The upper tail will then be =.2033, s the prbability f a stckut is.2033.

5 KEY TERMS Prbability density functin Unifrm prbability distributin Nrmal prbability distributin A functin used t cmpute prbabilities fr a cntinuus randm variable. The area under the graph f a prbability density functin ver an interval represents prbability. A cntinuus prbability distributin fr which the prbability that the randm variable will assume a value in any interval is the same fr each interval f equal length. A cntinuus prbability distributin. Its prbability density functin is bell-shaped and determined by its mean µ and standard deviatin σ. Standard nrmal prbability distributin A nrmal distributin with a mean f zer and a standard deviatin f ne. Cntinuity crrectin factr Expnential prbability distributin A value f.5 that is added t r subtracted frm a value f x when the cntinuus nrmal distributin is used t apprximate the discrete binmial distributin. A cntinuus prbability distributin that is useful in cmputing prbabilities fr the time it takes t cmplete a task. Key Frmulas fr Chapter 6: