Continuous Variables and Their Probability Distributions

Transcription

1 CHAPTER 4 Continuous Vriles nd Their Proility Distriutions 4. Introduction 4.2 The Proility Distriution for Continuous Rndom Vrile 4.3 Expected Vlues for Continuous Rndom Vriles 4.4 The Uniform Proility Distriution 4.5 The Norml Proility Distriution 4.6 The Gmm Proility Distriution 4.7 The Bet Proility Distriution 4.8 Some Generl Comments 4.9 Other Expected Vlues 4. Tcheysheff s Theorem 4. Expecttions of Discontinuous Functions nd Mixed Proility Distriutions (Optionl) 4.2 Summry References nd Further Redings 4. Introduction A moment of reflection on rndom vriles encountered in the rel world should convince you tht not ll rndom vriles of interest re discrete rndom vriles. The numer of dys tht it rins in period of n dys is discrete rndom vrile ecuse the numer of dys must tke one of the n + vlues,, 2,..., or n. Now consider the dily rinfll t specified geogrphicl point. Theoreticlly, with mesuring equipment of perfect ccurcy, the mount of rinfll could tke on ny vlue etween nd 5 inches. As result, ech of the uncountly infinite numer of points in the intervl (, 5) represents distinct possile vlue of the mount of 57

2 58 Chpter 4 Continuous Vriles nd Their Proility Distriutions rinfll in dy. A rndom vrile tht cn tke on ny vlue in n intervl is clled continuous, nd the purpose of this chpter is to study proility distriutions for continuous rndom vriles. The yield of n ntiiotic in fermenttion process is continuous rndom vrile, s is the length of life, in yers, of wshing mchine. The line segments over which these two rndom vriles re defined re contined in the positive hlf of the rel line. This does not men tht, if we oserved enough wshing mchines, we would eventully oserve n outcome corresponding to every vlue in the intervl (3, 7); rther it mens tht no vlue etween 3 nd 7 cn e ruled out s s possile vlue for the numer of yers tht wshing mchine remins in service. The proility distriution for discrete rndom vrile cn lwys e given y ssigning nonnegtive proility to ech of the possile vlues the vrile my ssume. In every cse, of course, the sum of ll the proilities tht we ssign must e equl to. Unfortuntely, the proility distriution for continuous rndom vrile cnnot e specified in the sme wy. It is mthemticlly impossile to ssign nonzero proilities to ll the points on line intervl while stisfying the requirement tht the proilities of the distinct possile vlues sum to. As result, we must develop different method to descrie the proility distriution for continuous rndom vrile. 4.2 The Proility Distriution for Continuous Rndom Vrile Before we cn stte forml definition for continuous rndom vrile, we must define the distriution function (or cumultive distriution function) ssocited with rndom vrile. DEFINITION 4. Let Y denote ny rndom vrile. The distriution function of Y, denoted y F(y), is such tht F(y) = P(Y y) for < y <. The nture of the distriution function ssocited with rndom vrile determines whether the vrile is continuous or discrete. Consequently, we will commence our discussion y exmining the distriution function for discrete rndom vrile nd noting the chrcteristics of this function. EXAMPLE 4. Solution Suppose tht Y hs inomil distriution with n = 2 nd p = /2. Find F(y). The proility function for Y is given y ( )( ) 2 y ( ) 2 y p(y) =, y =,, 2, y 2 2 which yields p() = /4, p() = /2, p(2) = /4.

3 4.2 The Proility Distriution for Continuous Rndom Vrile 59 FIGURE 4. Binomil distriution function, n = 2, p = /2 F( y) 3/4 /2 /4 2 y Wht is F( 2) = P(Y 2)? Becuse the only vlues of Y tht re ssigned positive proilities re,, nd 2 nd none of these vlues re less thn or equl to 2, F( 2) =. Using similr logic, F(y) = for ll y <. Wht is F(.5)? The only vlues of Y tht re less thn or equl to.5 nd hve nonzero proilities re the vlues nd. Therefore, F(.5) = P(Y.5) = P(Y = ) + P(Y = ) = (/4) + (/2) = 3/4. In generl,, for y <, /4, for y <, F(y) = P(Y y) = 3/4, for y < 2,, for y 2. A grph of F(y) is given in Figure 4.. In Exmple 4. the points etween nd or etween nd 2 ll hd proility nd contriuted nothing to the cumultive proility depicted y the distriution function. As result, the cumultive distriution function styed flt etween the possile vlues of Y nd incresed in jumps or steps t ech of the possile vlues of Y. Functions tht ehve in such mnner re clled step functions. Distriution functions for discrete rndom vriles re lwys step functions ecuse the cumultive distriution function increses only t the finite or countle numer of points with positive proilities. Becuse the distriution function ssocited with ny rndom vrile is such tht F(y) = P(Y y), from prcticl point of view it is cler tht F() = lim y P(Y y) must equl zero. If we consider ny two vlues y < y 2, then P(Y y ) P(Y y 2 ) tht is, F(y ) F(y 2 ). So, distriution function, F(y), is lwys monotonic, nondecresing function. Further, it is cler tht F( ) = lim y P(Y y) =. These three chrcteristics define the properties of ny distriution function nd re summrized in the following theorem.

4 6 Chpter 4 Continuous Vriles nd Their Proility Distriutions THEOREM 4. Properties of Distriution Function If F(y) is distriution function, then. F() lim F(y) =. y F(y) =. 2. F( ) lim y 3. F(y) is nondecresing function of y. [If y nd y 2 re ny vlues such tht y < y 2, then F(y ) F(y 2 ).] You should check tht the distriution function developed in Exmple 4. hs ech of these properties. Let us now exmine the distriution function for continuous rndom vrile. Suppose tht, for ll prcticl purposes, the mount of dily rinfll, Y, must e less thn 6 inches. For every y < y 2 6, the intervl (y, y 2 ) hs positive proility of including Y, no mtter how close y gets to y 2. It follows tht F(y) in this cse should e smooth, incresing function over some intervl of rel numers, s grphed in Figure 4.2. We re thus led to the definition of continuous rndom vrile. DEFINITION 4.2 A rndom vrile Y with distriution function F(y) is sid to e continuous if F(y) is continuous, for < y <. 2 FIGURE 4.2 Distriution function for continuous rndom vrile F( y) F ( y 2 ) F ( y ) y y 2 y. To e mthemticlly rigorous, if F(y) is vlid distriution function, then F(y) lso must e right continuous. 2. To e mthemticlly precise, we lso need the first derivtive of F(y) to exist nd e continuous except for, t most, finite numer of points in ny finite intervl. The distriution functions for the continuous rndom vriles discussed in this text stisfy this requirement.

5 4.2 The Proility Distriution for Continuous Rndom Vrile 6 If Y is continuous rndom vrile, then for ny rel numer y, P(Y = y) =. If this were not true nd P(Y = y ) = p >, then F(y) would hve discontinuity (jump) of size p t the point y, violting the ssumption tht Y ws continuous. Prcticlly speking, the fct tht continuous rndom vriles hve zero proility t discrete points should not other us. Consider the exmple of mesuring dily rinfll. Wht is the proility tht we will see dily rinfll mesurement of exctly 2.93 inches? It is quite likely tht we would never oserve tht exct vlue even if we took rinfll mesurements for lifetime, lthough we might see mny dys with mesurements etween 2 nd 3 inches. The derivtive of F(y) is nother function of prime importnce in proility theory nd sttistics. DEFINITION 4.3 Let F(y) e the distriution function for continuous rndom vrile Y. Then f (y), given y f (y) = df(y) = F (y) dy wherever the derivtive exists, is clled the proility density function for the rndom vrile Y. It follows from Definitions 4.2 nd 4.3 tht F(y) cn e written s F(y) = y f (t) dt, where f ( ) is the proility density function nd t is used s the vrile of integrtion. The reltionship etween the distriution nd density functions is shown grphiclly in Figure 4.3. The proility density function is theoreticl model for the frequency distriution (histogrm) of popultion of mesurements. For exmple, oservtions of the lengths of life of wshers of prticulr rnd will generte mesurements tht cn e chrcterized y reltive frequency histogrm, s discussed in Chpter. Conceptully, the experiment could e repeted d infinitum, therey generting reltive frequency distriution ( smooth curve) tht would chrcterize the popultion of interest to the mnufcturer. This theoreticl reltive frequency distriution corresponds to the proility density function for the length of life of single mchine, Y. FIGURE 4.3 The distriution function f ( y) F ( y ) y y

6 62 Chpter 4 Continuous Vriles nd Their Proility Distriutions Becuse the distriution function F(y) for ny rndom vrile lwys hs the properties given in Theorem 4., density functions must hve some corresponding properties. Becuse F(y) is nondecresing function, the derivtive f (y) is never negtive. Further, we know tht F( ) = nd, therefore, tht f (t) dt =. In summry, the properties of proility density function re s given in the following theorem. THEOREM 4.2 Properties of Density Function If f (y) is density function for continuous rndom vrile, then. f (y) for ll y, < y <. 2. f (y) dy =. The next exmple gives the distriution function nd density function for continuous rndom vrile. EXAMPLE 4.2 Solution Suppose tht, for y <, F(y) = y, for y,, for y >. Find the proility density function for Y nd grph it. Becuse the density function f (y) is the derivtive of the distriution function F(y), when the derivtive exists, d() =, for y <, dy f (y) = df(y) d(y) = =, for < y <, dy dy d() =, for y >, dy nd f (y) is undefined t y = nd y =. A grph of F(y) is shown in Figure 4.4. FIGURE 4.4 Distriution function F (y) for Exmple 4.2 F( y) y The grph of f (y) for Exmple 4.2 is shown in Figure 4.5. Notice tht the distriution nd density functions given in Exmple 4.2 hve ll the properties required

7 4.2 The Proility Distriution for Continuous Rndom Vrile 63 FIGURE 4.5 Density function f (y) for Exmple 4.2 f (y) y of distriution nd density functions, respectively. Moreover, F(y) is continuous function of y,ut f (y) is discontinuous t the points y =,. In generl, the distriution function for continuous rndom vrile must e continuous, ut the density function need not e everywhere continuous. EXAMPLE 4.3 Solution Let Y e continuous rndom vrile with proility density function given y { 3y 2, y, f (y) =, elsewhere. Find F(y). Grph oth f (y) nd F(y). The grph of f (y) ppers in Figure 4.6. Becuse F(y) = y f (t) dt, we hve, for this exmple, y dt =, for y <, F(y) = dt + y 3t 2 dt = + t 3] y = y3, for y, dt + 3t 2 dt + y dt = + t 3] + =, for < y. Notice tht some of the integrls tht we evluted yield vlue of. These re included for completeness in this initil exmple. In future clcultions, we will not explicitly disply ny integrl tht hs vlue. The grph of F(y) is given in Figure 4.7. FIGURE 4.6 Density function for Exmple 4.3 f (y) 3 2 y F(y ) gives the proility tht Y y. As you will see in susequent chpters, it is often of interest to determine the vlue, y, of rndom vrile Y tht is such tht P(Y y) equls or exceeds some specified vlue.

8 64 Chpter 4 Continuous Vriles nd Their Proility Distriutions FIGURE 4.7 Distriution function for Exmple 4.3 F( y) y DEFINITION 4.4 Let Y denote ny rndom vrile. If < p <, the pth quntile of Y, denoted y φ p, is the smllest vlue such tht P(Y φ q ) = F(φ p ) p. IfY is continuous, φ p is the smllest vlue such tht F(φ p ) = P(Y φ p ) = p. Some prefer to cll φ p the pth percentile of Y. An importnt specil cse is p = /2, nd φ.5 is the medin of the rndom vrile Y. In Exmple 4.3, the medin of the rndom vrile is such tht F(φ.5 ) =.5 nd is esily seen to e such tht (φ.5 ) 3 =.5, or equivlently, tht the medin of Y is φ.5 = (.5) /3 = The next step is to find the proility tht Y flls in specific intervl; tht is, P( Y ). From Chpter we know tht this proility corresponds to the re under the frequency distriution over the intervl y. Becuse f (y) is the theoreticl counterprt of the frequency distriution, we would expect P( Y ) to equl corresponding re under the density function f (y). This indeed is true ecuse, if <, P( < Y ) = P(Y ) P(Y ) = F() F() = Becuse P(Y = ) =, we hve the following result. f (y) dy. THEOREM 4.3 If the rndom vrile Y hs density function f (y) nd <, then the proility tht Y flls in the intervl [, ]is P( Y ) = f (y) dy. This proility is the shded re in Figure 4.8. FIGURE 4.8 P ( Y ) f (y) y

9 4.2 The Proility Distriution for Continuous Rndom Vrile 65 If Y is continuous rndom vrile nd nd re constnts such tht <, then P(Y = ) = nd P(Y = ) = nd Theorem 4.3 implies tht P( < Y < ) = P( Y < ) = P( < Y ) = P( Y ) = f (y) dy. The fct tht the ove string of equlities is not, in generl, true for discrete rndom vriles is illustrted in Exercise 4.7. EXAMPLE 4.4 Solution Given f (y) = cy 2, y 2, nd f (y) = elsewhere, find the vlue of c for which f (y) is vlid density function. We require vlue for c such tht F( ) = f (y) dy = = 2 ] 2 cy 2 dy = cy3 = 3 ( ) 8 c. 3 Thus, (8/3)c =, nd we find tht c = 3/8. EXAMPLE 4.5 Find P( Y 2) for Exmple 4.4. Also find P( < Y < 2). Solution P( Y 2) = 2 f (y) dy = y 2 dy = ( ) 3 y ] 2 = 7 8. Becuse Y hs continuous distriution, it follows tht P(Y = ) = P(Y = 2) = nd, therefore, tht P( < Y < 2) = P( Y 2) = y 2 dy = 7 8. Proility sttements regrding continuous rndom vrile Y re meningful only if, first, the integrl defining the proility exists nd, second, the resulting proilities gree with the xioms of Chpter 2. These two conditions will lwys e stisfied if we consider only proilities ssocited with finite or countle collection of intervls. Becuse we lmost lwys re interested in proilities tht continuous vriles fll in intervls, this considertion will cuse us no prcticl difficulty. Some density functions tht provide good models for popultion frequency distriutions encountered in prcticl pplictions re presented in susequent sections.

10 66 Chpter 4 Continuous Vriles nd Their Proility Distriutions Exercises 4. Let Y e rndom vrile with p(y) given in the tle elow. y p(y) Give the distriution function, F(y). Be sure to specify the vlue of F(y) for ll y, < y <. Sketch the distriution function given in prt (). 4.2 A ox contins five keys, only one of which will open lock. Keys re rndomly selected nd tried, one t time, until the lock is opened (keys tht do not work re discrded efore nother is tried). Let Y e the numer of the tril on which the lock is opened. Find the proility function for Y. Give the corresponding distriution function. c Wht is P(Y < 3)? P(Y 3)? P(Y = 3)? d If Y is continuous rndom vrile, we rgued tht, for ll < <, P(Y = ) =. Do ny of your nswers in prt (c) contrdict this clim? Why? 4.3 A Bernoulli rndom vrile is one tht ssumes only two vlues, nd with p() = p nd p() = p q. Sketch the corresponding distriution function. Show tht this distriution function hs the properties given in Theorem Let Y e inomil rndom vrile with n = nd success proility p. Find the proility nd distriution function for Y. Compre the distriution function from prt () with tht in Exercise 4.3(). Wht do you conclude? 4.5 Suppose tht Y is rndom vrile tht tkes on only integer vlues, 2,...nd hs distriution function F(y). Show tht the proility function p(y) = P(Y = y) is given y { F(), y =, p(y) = F(y) F(y ), y = 2, 3, Consider rndom vrile with geometric distriution (Section 3.5); tht is, p(y) = q y p, y =, 2, 3,..., < p <. Show tht Y hs distriution function F(y) such tht F(i) = q i, i =,, 2,...nd tht, in generl, {, y <, F(y) = q i, i y < i +, for i =,, 2,... Show tht the preceding cumultive distriution function hs the properties given in Theorem Let Y e inomil rndom vrile with n = nd p =.2. Use Tle, Appendix 3, to otin P(2 < Y < 5) nd P(2 Y < 5). Are the proilities tht Y flls in the intevls (2, 5) nd [2, 5) equl? Why or why not?

11 Exercises 67 Use Tle, Appendix 3, to otin P(2 < Y 5) nd P(2 Y 5). Are these two proilities equl? Why or why not? c Erlier in this section, we rgued tht if Y is continuous nd <, then P( < Y < ) = P( Y < ). Does the result in prt () contrdict this clim? Why? 4.8 Suppose tht Y hs density function { ky( y), y, f (y) =, elsewhere. Find the vlue of k tht mkes f (y) proility density function. Find P(.4 Y ). c Find P(.4 Y < ). d Find P(Y.4 Y.8). e Find P(Y <.4 Y <.8). 4.9 A rndom vrile Y hs the following distriution function:, for y < 2, /8, for 2 y < 2.5, 3/6, for 2.5 y < 4, F(y) = P(Y y) = /2 for 4 y < 5.5, 5/8, for 5.5 y < 6, /6, for 6 y < 7,, for y 7. Is Y continuous or discrete rndom vrile? Why? Wht vlues of Y re ssigned positive proilities? c Find the proility function for Y. d Wht is the medin, φ.5,ofy? 4. Refer to the density function given in Exercise 4.8. Find the.95-quntile, φ.95, such tht P(Y φ.95 ) =.95. Find vlue y so tht P(Y < y ) =.95. c Compre the vlues for φ.95 nd y tht you otined in prts () nd (). Explin the reltionship etween these two vlues. 4. Suppose tht Y possesses the density function { cy, y 2, f (y) =, elsewhere. Find the vlue of c tht mkes f (y) proility density function. Find F(y). c Grph f (y) nd F(y). d Use F(y) to find P( Y 2). e Use f (y) nd geometry to find P( Y 2). 4.2 The length of time to filure (in hundreds of hours) for trnsistor is rndom vrile Y with distriution function given y {, y <, F(y) = e y2, y.

12 68 Chpter 4 Continuous Vriles nd Their Proility Distriutions Show tht F(y) hs the properties of distriution function. Find the.3-quntile, φ.3,ofy. c Find f (y). d Find the proility tht the trnsistor opertes for t lest 2 hours. e Find P(Y > Y 2). 4.3 A supplier of kerosene hs 5-gllon tnk tht is filled t the eginning of ech week. His weekly demnd shows reltive frequency ehvior tht increses stedily up to gllons nd then levels off etween nd 5 gllons. If Y denotes weekly demnd in hundreds of gllons, the reltive frequency of demnd cn e modeled y y, y, f (y) =, < y.5,, elsewhere. Find F(y). Find P( Y.5). c Find P(.5 Y.2). 4.4 A gs sttion opertes two pumps, ech of which cn pump up to, gllons of gs in month. The totl mount of gs pumped t the sttion in month is rndom vrile Y (mesured in, gllons) with proility density function given y y, < y <, f (y) = 2 y, y < 2,, elsewhere. Grph f (y). Find F(y) nd grph it. c Find the proility tht the sttion will pump etween 8 nd 2, gllons in prticulr month. d Given tht the sttion pumped more thn, gllons in prticulr month, find the proility tht the sttion pumped more thn 5, gllons during the month. 4.5 As mesure of intelligence, mice re timed when going through mze to rech rewrd of food. The time (in seconds) required for ny mouse is rndom vrile Y with density function given y f (y) = y, y,, elsewhere, where is the minimum possile time needed to trverse the mze. Show tht f (y) hs the properties of density function. Find F(y). c Find P(Y > + c) for positive constnt c. d If c nd d re oth positive constnts such tht d > c, find P(Y > + d Y > + c). 4.6 Let Y possess density function { c(2 y), y 2, f (y) =, elsewhere.

13 Exercises 69 Find c. Find F(y). c Grph f (y) nd F(y). d Use F(y) in prt () to find P( Y 2). e Use geometry nd the grph for f (y) to clculte P( Y 2). 4.7 The length of time required y students to complete one-hour exm is rndom vrile with density function given y { cy 2 + y, y, f (y) =, elsewhere. Find c. Find F(y). c Grph f (y) nd F(y). d Use F(y) in prt () to find F( ), F(), nd F(). e Find the proility tht rndomly selected student will finish in less thn hlf n hour. f Given tht prticulr student needs t lest 5 minutes to complete the exm, find the proility tht she will require t lest 3 minutes to finish. 4.8 Let Y hve the density function given y.2, < y, f (y) =.2 + cy, < y,, elsewhere. Find c. Find F(y). c Grph f (y) nd F(y). d Use F(y) in prt () to find F( ), F(), nd F(). e Find P( Y.5). f Find P(Y >.5 Y >.). 4.9 Let the distriution function of rndom vrile Y e Find the density function of Y. Find P( Y 3). c Find P(Y.5). d Find P(Y Y 3)., y, y 8, < y < 2, F(y) = y 2 6, 2 y < 4,, y 4.

14 7 Chpter 4 Continuous Vriles nd Their Proility Distriutions 4.3 Expected Vlues for Continuous Rndom Vriles The next step in the study of continuous rndom vriles is to find their mens, vrinces, nd stndrd devitions, therey cquiring numericl descriptive mesures ssocited with their distriutions. Mny times it is difficult to find the proility distriution for rndom vrile Y or function of rndom vrile, g(y ). Even if the density function for rndom vrile is known, it cn e difficult to evlute pproprite integrls (we will see this to e the cse when rndom vrile hs gmm distriution, Section 4.6). When we encounter these situtions, the pproximte ehvior of vriles of interest cn e estlished y using their moments nd the empiricl rule or Tcheysheff s theorem (Chpters nd 3). DEFINITION 4.5 The expected vlue of continuous rndom vrile Y is E(Y ) = provided tht the integrl exists. 3 yf(y) dy, If the definition of the expected vlue for discrete rndom vrile Y, E(Y ) = y yp(y), is meningful, then Definition 4.4 lso should gree with our intuitive notion of men. The quntity f (y) dy corresponds to p(y) for the discrete cse, nd integrtion evolves from nd is nlogous to summtion. Hence, E(Y ) in Definition 4.5 grees with our notion of n verge, or men. As in the discrete cse, we re sometimes interested in the expected vlue of function of rndom vrile. A result tht permits us to evlute such n expected vlue is given in the following theorem. THEOREM 4.4 Let g(y ) e function of Y ; then the expected vlue of g(y ) is given y E [g(y )] = provided tht the integrl exists. g(y) f (y) dy, The proof of Theorem 4.4 is similr to tht of Theorem 3.2 nd is omitted. The expected vlues of three importnt functions of continuous rndom vrile Y evolve 3. Techniclly, E(Y ) is sid to exist if y f (y) dy <. This will e the cse in ll expecttions tht we discuss, nd we will not mention this dditionl condition ech time tht we define n expected vlue.

15 4.3 Expected Vlues for Continuous Rndom Vriles 7 s consequence of well-known theorems of integrtion. As expected, these results led to conclusions nlogous to those contined in Theorems 3.3, 3.4, nd 3.5. As consequence, the proof of Theorem 4.5 will e left s n exercise. THEOREM 4.5 Let c e constnt nd let g(y ), g (Y ), g 2 (Y ),...,g k (Y ) e functions of continuous rndom vrile Y. Then the following results hold:. E(c) = c. 2. E[cg(Y )] = ce[g(y )]. 3. E[g (Y )+g 2 (Y )+ +g k (Y )] = E[g (Y )]+E[g 2 (Y )]+ +E[g k (Y )]. As in the cse of discrete rndom vriles, we often seek the expected vlue of the function g(y ) = (Y μ) 2. As efore, the expected vlue of this function is the vrince of the rndom vrile Y. Tht is, s in Definition 3.5, V (Y ) = E(Y μ) 2. It is simple exercise to show tht Theorem 4.5 implies tht V (Y ) = E(Y 2 ) μ 2. EXAMPLE 4.6 In Exmple 4.4 we determined tht f (y) = (3/8)y 2 for y 2, f (y) = elsewhere, is vlid density function. If the rndom vrile Y hs this density function, find μ = E(Y ) nd σ 2 = V (Y ). Solution According to Definition 4.5, E(Y ) = = = 2 ( 3 8 yf(y) dy ( ) 3 y y 2 dy 8 )( ) ] 2 y 4 =.5. 4 The vrince of Y cn e found once we determine E(Y 2 ). In this cse, E(Y 2 ) = = = 2 ( 3 8 y 2 f (y) dy ( ) 3 y 2 y 2 dy 8 )( ) ] 2 y 5 = Thus, σ 2 = V (Y ) = E(Y 2 ) [E(Y )] 2 = 2.4 (.5) 2 =.5.

16 72 Chpter 4 Continuous Vriles nd Their Proility Distriutions Exercises 4.2 If, s in Exercise 4.6, Y hs density function { (/2)(2 y), y 2, f (y) =, elsewhere, find the men nd vrince of Y. 4.2 If, s in Exercise 4.7, Y hs density function { (3/2)y 2 + y, y, f (y) =, elsewhere, find the men nd vrince of Y If, s in Exercise 4.8, Y hs density function.2, < y, f (y) =.2 + (.2)y, < y,, elsewhere, find the men nd vrince of Y Prove Theorem If Y is continuous rndom vrile with density function f (y), use Theorem 4.5 to prove tht σ 2 = V (Y ) = E(Y 2 ) [E(Y )] If, s in Exercise 4.9, Y hs distriution function, y, y 8, < y < 2, F(y) = y 2 6, 2 y < 4,, y 4, find the men nd vrince of Y If Y is continuous rndom vrile with men μ nd vrince σ 2 nd nd re constnts, use Theorem 4.5 to prove the following: E(Y + ) = E(Y ) + = μ +. V (Y + ) = 2 V (Y ) = 2 σ For certin ore smples, the proportion Y of impurities per smple is rndom vrile with density function given in Exercise 4.2. The dollr vlue of ech smple is W = 5.5Y. Find the men nd vrince of W The proportion of time per dy tht ll checkout counters in supermrket re usy is rndom vrile Y with density function { cy 2 ( y) 4, y, f (y) =, elsewhere. Find the vlue of c tht mkes f (y) proility density function. Find E(Y ).

17 Exercises The temperture Y t which thermostticlly controlled switch turns on hs proility density function given y { /2, 59 y 6, f (y) =, elsewhere. Find E(Y ) nd V (Y ). 4.3 The proportion of time Y tht n industril root is in opertion during 4-hour week is rndom vrile with proility density function { 2y, y, f (y) =, elsewhere. Find E(Y ) nd V (Y ). For the root under study, the profit X for week is given y X = 2Y 6. Find E(X) nd V (X). c Find n intervl in which the profit should lie for t lest 75% of the weeks tht the root is in use. 4.3 The ph of wter smples from specific lke is rndom vrile Y with proility density function given y { (3/8)(7 y) 2, 5 y 7, f (y) =, elsewhere. Find E(Y ) nd V (Y ). Find n intervl shorter thn (5, 7) in which t lest three-fourths of the ph mesurements must lie. c Would you expect to see ph mesurement elow 5.5 very often? Why? 4.32 Weekly CPU time used y n ccounting firm hs proility density function (mesured in hours) given y { (3/64)y 2 (4 y), y 4, f (y) =, elsewhere. Find the expected vlue nd vrince of weekly CPU time. The CPU time costs the firm $2 per hour. Find the expected vlue nd vrince of the weekly cost for CPU time. c Would you expect the weekly cost to exceed $6 very often? Why? 4.33 Dily totl solr rdition for specified loction in Florid in Octoer hs proility density function given y { (3/32)(y 2)(6 y), 2 y 6, f (y) =, elsewhere, with mesurements in hundreds of clories. Find the expected dily solr rdition for Octoer. *4.34 Suppose tht Y is continuous rndom vrile with density f (y) tht is positive only if y. If F(y) is the distriution function, show tht E(Y ) = yf(y) dy = [ F(y)] dy. [Hint: If y >, y = y dt, nd E(Y ) = yf(y) dy = { y dt} f (y) dy. Exchnge the order of integrtion to otin the desired result.] 4 4. Exercises preceded y n sterisk re optionl.

18 74 Chpter 4 Continuous Vriles nd Their Proility Distriutions *4.35 If Y is continuous rndom vrile such tht E[(Y ) 2 ] < for ll, show tht E[(Y ) 2 ] is minimized when = E(Y ).[Hint: E[(Y ) 2 ] = E({[Y E(Y )] + [E(Y ) ]} 2 ).] *4.36 Is the result otined in Exercise 4.35 lso vlid for discrete rndom vriles? Why? *4.37 If Y is continuous rndom vrile with density function f (y) tht is symmetric out (tht is, f (y) = f ( y) for ll y) nd E(Y ) exists, show tht E(Y ) =. [Hint: E(Y ) = yf(y) dy + yf(y) dy. Mke the chnge of vrile w = y in the first integrl.] 4.4 The Uniform Proility Distriution Suppose tht us lwys rrives t prticulr stop etween 8: nd 8: A.M. nd tht the proility tht the us will rrive in ny given suintervl of time is proportionl only to the length of the suintervl. Tht is, the us is s likely to rrive etween 8: nd 8:2 s it is to rrive etween 8:6 nd 8:8. Let Y denote the length of time person must wit for the us if tht person rrived t the us stop t exctly 8:. If we crefully mesured in minutes how long fter 8: the us rrived for severl mornings, we could develop reltive frequency histogrm for the dt. From the description just given, it should e cler tht the reltive frequency with which we oserved vlue of Y etween nd 2 would e pproximtely the sme s the reltive frequency with which we oserved vlue of Y etween 6 nd 8. A resonle model for the density function of Y is given in Figure 4.9. Becuse res under curves represent proilities for continuous rndom vriles nd A = A 2 (y inspection), it follows tht P( Y 2) = P(6 Y 8), s desired. The rndom vrile Y just discussed is n exmple of rndom vrile tht hs uniform distriution. The generl form for the density function of rndom vrile with uniform distriution is s follows. DEFINITION 4.6 If θ <θ 2, rndom vrile Y is sid to hve continuous uniform proility distriution on the intervl (θ,θ 2 ) if nd only if the density function of Y is, θ y θ 2, f (y) = θ 2 θ, elsewhere. FIGURE 4.9 Density function for Y f ( y) A A y

19 4.4 The Uniform Proility Distriution 75 In the us prolem we cn tke θ = nd θ 2 = ecuse we re interested only in prticulr ten-minute intervl. The density function discussed in Exmple 4.2 is uniform distriution with θ = nd θ 2 =. Grphs of the distriution function nd density function for the rndom vrile in Exmple 4.2 re given in Figures 4.4 nd 4.5, respectively. DEFINITION 4.7 The constnts tht determine the specific form of density function re clled prmeters of the density function. The quntities θ nd θ 2 re prmeters of the uniform density function nd re clerly meningful numericl vlues ssocited with the theoreticl density function. Both the rnge nd the proility tht Y will fll in ny given intervl depend on the vlues of θ nd θ 2. Some continuous rndom vriles in the physicl, mngement, nd iologicl sciences hve pproximtely uniform proility distriutions. For exmple, suppose tht the numer of events, such s clls coming into switchord, tht occur in the time intervl (, t) hs Poisson distriution. If it is known tht exctly one such event hs occurred in the intervl (, t), then the ctul time of occurrence is distriuted uniformly over this intervl. EXAMPLE 4.7 Solution Arrivls of customers t checkout counter follow Poisson distriution. It is known tht, during given 3-minute period, one customer rrived t the counter. Find the proility tht the customer rrived during the lst 5 minutes of the 3-minute period. As just mentioned, the ctul time of rrivl follows uniform distriution over the intervl of (, 3). IfY denotes the rrivl time, then P(25 Y 3) = dy = 3 3 = 5 3 = 6. The proility of the rrivl occurring in ny other 5-minute intervl is lso /6. As we will see, the uniform distriution is very importnt for theoreticl resons. Simultion studies re vlule techniques for vlidting models in sttistics. If we desire set of oservtions on rndom vrile Y with distriution function F(y), we often cn otin the desired results y trnsforming set of oservtions on uniform rndom vrile. For this reson most computer systems contin rndom numer genertor tht genertes oserved vlues for rndom vrile tht hs continuous uniform distriution.

20 76 Chpter 4 Continuous Vriles nd Their Proility Distriutions THEOREM 4.6 If θ <θ 2 nd Y is rndom vrile uniformly distriuted on the intervl (θ,θ 2 ), then μ = E(Y ) = θ + θ 2 2 Proof By Definition 4.5, E(Y ) = nd σ 2 = V (Y ) = (θ 2 θ ) 2. 2 yf(y) dy θ2 ( ) = y dy θ θ 2 θ ( ) y 2 ] θ2 = = θ θ θ 2 θ 2 θ 2(θ 2 θ ) = θ 2 + θ. 2 Note tht the men of uniform rndom vrile is simply the vlue midwy etween the two prmeter vlues, θ nd θ 2. The derivtion of the vrince is left s n exercise. Exercises 4.38 Suppose tht Y hs uniform distriution over the intervl (, ). Find F(y). Show tht P( Y + ), for,, nd + depends only upon the vlue of If prchutist lnds t rndom point on line etween mrkers A nd B, find the proility tht she is closer to A thn to B. Find the proility tht her distnce to A is more thn three times her distnce to B. 4.4 Suppose tht three prchutists operte independently s descried in Exercise Wht is the proility tht exctly one of the three lnds pst the midpoint etween A nd B? 4.4 A rndom vrile Y hs uniform distriution over the intervl (θ,θ 2 ). Derive the vrince of Y The medin of the distriution of continuous rndom vrile Y is the vlue φ.5 such tht P(Y φ.5 ) =.5. Wht is the medin of the uniform distriution on the intervl (θ,θ 2 )? 4.43 A circle of rdius r hs re A = πr 2. If rndom circle hs rdius tht is uniformly distriuted on the intervl (, ), wht re the men nd vrince of the re of the circle? 4.44 The chnge in depth of river from one dy to the next, mesured (in feet) t specific loction, is rndom vrile Y with the following density function: { k, 2 y 2 f (y) =, elsewhere.

21 Exercises 77 Determine the vlue of k. Otin the distriution function for Y Upon studying low ids for shipping contrcts, microcomputer mnufcturing compny finds tht intrstte contrcts hve low ids tht re uniformly distriuted etween 2 nd 25, in units of thousnds of dollrs. Find the proility tht the low id on the next intrstte shipping contrct is elow $22,. is in excess of $24, Refer to Exercise Find the expected vlue of low ids on contrcts of the type descried there The filure of circuit ord interrupts work tht utilizes computing system until new ord is delivered. The delivery time, Y, is uniformly distriuted on the intervl one to five dys. The cost of ord filure nd interruption includes the fixed cost c of new ord nd cost tht increses proportionlly to Y 2.IfC is the cost incurred, C = c + c Y 2. Find the proility tht the delivery time exceeds two dys. In terms of c nd c, find the expected cost ssocited with single filed circuit ord Beginning t 2: midnight, computer center is up for one hour nd then down for two hours on regulr cycle. A person who is unwre of this schedule dils the center t rndom time etween 2: midnight nd 5: A.M. Wht is the proility tht the center is up when the person s cll comes in? 4.49 A telephone cll rrived t switchord t rndom within one-minute intervl. The switch ord ws fully usy for 5 seconds into this one-minute period. Wht is the proility tht the cll rrived when the switchord ws not fully usy? 4.5 If point is rndomly locted in n intervl (, ) nd if Y denotes the loction of the point, then Y is ssumed to hve uniform distriution over (, ). A plnt efficiency expert rndomly selects loction long 5-foot ssemly line from which to oserve the work hits of the workers on the line. Wht is the proility tht the point she selects is c within 25 feet of the end of the line? within 25 feet of the eginning of the line? closer to the eginning of the line thn to the end of the line? 4.5 The cycle time for trucks huling concrete to highwy construction site is uniformly distriuted over the intervl 5 to 7 minutes. Wht is the proility tht the cycle time exceeds 65 minutes if it is known tht the cycle time exceeds 55 minutes? 4.52 Refer to Exercise 4.5. Find the men nd vrince of the cycle times for the trucks The numer of defective circuit ords coming off soldering mchine follows Poisson distriution. During specific eight-hour dy, one defective circuit ord ws found. c Find the proility tht it ws produced during the first hour of opertion during tht dy. Find the proility tht it ws produced during the lst hour of opertion during tht dy. Given tht no defective circuit ords were produced during the first four hours of opertion, find the proility tht the defective ord ws mnufctured during the fifth hour In using the tringultion method to determine the rnge of n coustic source, the test equipment must ccurtely mesure the time t which the sphericl wve front rrives t receiving

22 78 Chpter 4 Continuous Vriles nd Their Proility Distriutions sensor. According to Perruzzi nd Hillird (984), mesurement errors in these times cn e modeled s possessing uniform distriution from.5 to +.5 μs (microseconds). Wht is the proility tht prticulr rrivl-time mesurement will e ccurte to within. μs? Find the men nd vrince of the mesurement errors Refer to Exercise Suppose tht mesurement errors re uniformly distriuted etween.2 to +.5 μs. Wht is the proility tht prticulr rrivl-time mesurement will e ccurte to within. μs? Find the men nd vrince of the mesurement errors Refer to Exmple 4.7. Find the conditionl proility tht customer rrives during the lst 5 minutes of the 3-minute period if it is known tht no one rrives during the first minutes of the period According to Zimmels (983), the sizes of prticles used in sedimenttion experiments often hve uniform distriution. In sedimenttion involving mixtures of prticles of vrious sizes, the lrger prticles hinder the movements of the smller ones. Thus, it is importnt to study oth the men nd the vrince of prticle sizes. Suppose tht sphericl prticles hve dimeters tht re uniformly distriuted etween. nd.5 centimeters. Find the men nd vrince of the volumes of these prticles. (Recll tht the volume of sphere is (4/3)πr 3.) 4.5 The Norml Proility Distriution The most widely used continuous proility distriution is the norml distriution, distriution with the fmilir ell shpe tht ws discussed in connection with the empiricl rule. The exmples nd exercises in this section illustrte some of the mny rndom vriles tht hve distriutions tht re closely pproximted y norml proility distriution. In Chpter 7 we will present n rgument tht t lest prtilly explins the common occurrence of norml distriutions of dt in nture. The norml density function is s follows: DEFINITION 4.8 A rndom vrile Y is sid to hve norml proility distriution if nd only if, for σ>nd <μ<, the density function of Y is f (y) = σ /(2σ 2), < y <. 2π e (y μ)2 Notice tht the norml density function contins two prmeters, μ nd σ. THEOREM 4.7 If Y is normlly distriuted rndom vrile with prmeters μ nd σ, then E(Y ) = μ nd V (Y ) = σ 2.

23 4.5 The Norml Proility Distriution 79 FIGURE 4. The norml proility density function f (y) y The proof of this theorem will e deferred to Section 4.9, where we derive the moment-generting function of normlly distriuted rndom vrile. The results contined in Theorem 4.7 imply tht the prmeter μ loctes the center of the distriution nd tht σ mesures its spred. A grph of norml density function is shown in Figure 4.. Ares under the norml density function corresponding to P( Y ) require evlution of the integrl σ 2π e (y μ)2 /(2σ 2) dy. Unfortuntely, closed-form expression for this integrl does not exist; hence, its evlution requires the use of numericl integrtion techniques. Proilities nd quntiles for rndom vriles with norml distriutions re esily found using R nd S-Plus. If Y hs norml distriution with men μ nd stndrd devition σ, the R (or S-Plus) commnd pnorm(y,μ,σ ) genertes P(Y y ) wheres qnorm(p,μ,σ ) yields the pth quntile, the vlue of φ p such tht P(Y φ p ) = p. Although there re infinitely mny norml distriutions (μ cn tke on ny finite vlue, wheres σ cn ssume ny positive finite vlue), we need only one tle Tle 4, Appendix 3 to compute res under norml densities. Proilities nd quntiles ssocited with normlly distriuted rndom vriles cn lso e found using the pplet Norml Til Ares nd Quntiles ccessile t wckerly. The only rel enefit ssocited with using softwre to otin proilities nd quntiles ssocited with normlly distriuted rndom vriles is tht the softwre provides nswers tht re correct to greter numer of deciml plces. The norml density function is symmetric round the vlue μ, so res need e tulted on only one side of the men. The tulted res re to the right of points z, where z is the distnce from the men, mesured in stndrd devitions. This re is shded in Figure 4.. EXAMPLE 4.8 Let Z denote norml rndom vrile with men nd stndrd devition. Find P(Z > 2). Find P( 2 Z 2). c Find P( Z.73).

24 8 Chpter 4 Continuous Vriles nd Their Proility Distriutions FIGURE 4. Tulted re for the norml density function f (y) z + z y Solution Since μ = nd σ =, the vlue 2 is ctully z = 2 stndrd devitions ove the men. Proceed down the first (z) column in Tle 4, Appendix 3, nd red the re opposite z = 2.. This re, denoted y the symol A(z), is A(2.) =.228. Thus, P(Z > 2) =.228. Refer to Figure 4.2, where we hve shded the re of interest. In prt () we determined tht A = A(2.) =.228. Becuse the density function is symmetric out the men μ =, it follows tht A 2 = A =.228 nd hence tht P( 2 Z 2) = A A 2 = 2(.228) = c Becuse P(Z > ) = A() =.5, we otin tht P( Z.73) =.5 A(.73), where A(.73) is otined y proceeding down the z column in Tle 4, Appendix 3, to the entry.7 nd then cross the top of the tle to the column leled.3 to red A(.73) =.48. Thus, P( Z.73) =.5.48 = FIGURE 4.2 Desired re for Exmple 4.8() A A y EXAMPLE 4.9 Solution The chievement scores for college entrnce exmintion re normlly distriuted with men 75 nd stndrd devition. Wht frction of the scores lies etween 8 nd 9? Recll tht z is the distnce from the men of norml distriution expressed in units of stndrd devition. Thus, z = y μ. σ

25 Exercises 8 FIGURE 4.3 Required re for Exmple 4.9 A.5.5 z Then the desired frction of the popultion is given y the re etween z = =.5 nd z 2 = =.5. This re is shded in Figure 4.3. You cn see from Figure 4.3 tht A = A(.5) A(.5) = =.247. We cn lwys trnsform norml rndom vrile Y to stndrd norml rndom vrile Z y using the reltionship Z = Y μ. σ Tle 4, Appendix 3, cn then e used to compute proilities, s shown here. Z loctes point mesured from the men of norml rndom vrile, with the distnce expressed in units of the stndrd devition of the originl norml rndom vrile. Thus, the men vlue of Z must e, nd its stndrd devition must equl. The proof tht the stndrd norml rndom vrile, Z, is normlly distriuted with men nd stndrd devition is given in Chpter 6. The pplet Norml Proilities, ccessile t wckerly, illustrtes the correspondence etween norml proilities on the originl nd trnsformed (z) scles. To nswer the question posed in Exmple 4.9, locte the intervl of interest, (8, 9), on the lower horizontl xis leled Y. The corresponding z-scores re given on the upper horizontl xis, nd it is cler tht the shded re gives P(8 < Y < 9) = P(.5 < Z <.5) =.247 (see Figure 4.4). A few of the exercises t the end of this section suggest tht you use this pplet to reinforce the clcultions of proilities ssocited with normlly distriuted rndom vriles. Exercises 4.58 Use Tle 4, Appendix 3, to find the following proilities for stndrd norml rndom vrile Z: P( Z.2) P(.9 Z ) c P(.3 Z.56)

26 82 Chpter 4 Continuous Vriles nd Their Proility Distriutions FIGURE 4.4 Required re for Exmple 4.9, using oth the originl nd trnsformed (z) scles P(8. < Y < 9.) = P(.5 < Z <.5) = Pro = Z Y d P(.2 Z.2) e P(.56 Z.2) f Applet Exercise Use the pplet Norml Proilities to otin P( Z.2). Why re the vlues given on the two horizontl xes identicl? 4.59 If Z is stndrd norml rndom vrile, find the vlue z such tht P(Z > z ) =.5. P(Z < z ) = c P( z < Z < z ) =.9. d P( z < Z < z ) = A normlly distriuted rndom vrile hs density function f (y) = σ 2π e (y μ)2 /(2σ 2), < y <. Using the fundmentl properties ssocited with ny density function, rgue tht the prmeter σ must e such tht σ>. 4.6 Wht is the medin of normlly distriuted rndom vrile with men μ nd stndrd devition σ? 4.62 If Z is stndrd norml rndom vrile, wht is P(Z 2 < )? P(Z 2 < )? 4.63 A compny tht mnufctures nd ottles pple juice uses mchine tht utomticlly fills 6-ounce ottles. There is some vrition, however, in the mounts of liquid dispensed into the ottles tht re filled. The mount dispensed hs een oserved to e pproximtely normlly distriuted with men 6 ounces nd stndrd devition ounce.

27 Exercises 83 Use Tle 4, Appendix 3, to determine the proportion of ottles tht will hve more thn 7 ounces dispensed into them. Applet Exercise Use the pplet Norml Proilities to otin the nswer to prt () The weekly mount of money spent on mintennce nd repirs y compny ws oserved, over long period of time, to e pproximtely normlly distriuted with men $4 nd stndrd devition $2. If $45 is udgeted for next week, wht is the proility tht the ctul costs will exceed the udgeted mount? Answer the question, using Tle 4, Appendix 3. Applet Exercise Use the pplet Norml Proilities to otin the nswer. c Why re the leled vlues different on the two horizontl xes? 4.65 In Exercise 4.64, how much should e udgeted for weekly repirs nd mintennce to provide tht the proility the udgeted mount will e exceeded in given week is only.? 4.66 A mchining opertion produces erings with dimeters tht re normlly distriuted with men 3.5 inches nd stndrd devition. inch. Specifictions require the ering dimeters to lie in the intervl 3. ±.2 inches. Those outside the intervl re considered scrp nd must e remchined. With the existing mchine setting, wht frction of totl production will e scrp? Answer the question, using Tle 4, Appendix 3. Applet Exercise Otin the nswer, using the pplet Norml Proilities In Exercise 4.66, wht should the men dimeter e in order tht the frction of erings scrpped e minimized? 4.68 The grde point verges (GPAs) of lrge popultion of college students re pproximtely normlly distriuted with men 2.4 nd stndrd devition.8. Wht frction of the students will possess GPA in excess of 3.? Answer the question, using Tle 4, Appendix 3. Applet Exercise Otin the nswer, using the pplet Norml Til Ares nd Quntiles Refer to Exercise If students possessing GPA less thn.9 re dropped from college, wht percentge of the students will e dropped? 4.7 Refer to Exercise Suppose tht three students re rndomly selected from the student ody. Wht is the proility tht ll three will possess GPA in excess of 3.? 4.7 Wires mnufctured for use in computer system re specified to hve resistnces etween.2 nd.4 ohms. The ctul mesured resistnces of the wires produced y compny A hve norml proility distriution with men.3 ohm nd stndrd devition.5 ohm. Wht is the proility tht rndomly selected wire from compny A s production will meet the specifictions? If four of these wires re used in ech computer system nd ll re selected from compny A, wht is the proility tht ll four in rndomly selected system will meet the specifictions? 4.72 One method of rriving t economic forecsts is to use consensus pproch. A forecst is otined from ech of lrge numer of nlysts; the verge of these individul forecsts is the consensus forecst. Suppose tht the individul 996 Jnury prime interest rte forecsts of ll economic nlysts re pproximtely normlly distriuted with men 7% nd stndrd

28 84 Chpter 4 Continuous Vriles nd Their Proility Distriutions devition 2.6%. If single nlyst is rndomly selected from mong this group, wht is the proility tht the nlyst s forecst of the prime interest rte will exceed %? e less thn 9%? 4.73 The width of olts of fric is normlly distriuted with men 95 mm (millimeters) nd stndrd devition mm. Wht is the proility tht rndomly chosen olt hs width of etween 947 nd 958 mm? Wht is the pproprite vlue for C such tht rndomly chosen olt hs width less thn C with proility.853? 4.74 Scores on n exmintion re ssumed to e normlly distriuted with men 78 nd vrince 36. Wht is the proility tht person tking the exmintion scores higher thn 72? Suppose tht students scoring in the top % of this distriution re to receive n A grde. Wht is the minimum score student must chieve to ern n A grde? c Wht must e the cutoff point for pssing the exmintion if the exminer wnts only the top 28.% of ll scores to e pssing? d Approximtely wht proportion of students hve scores 5 or more points ove the score tht cuts off the lowest 25%? e Applet Exercise Answer prts () (d), using the pplet Norml Til Ares nd Quntiles. f If it is known tht student s score exceeds 72, wht is the proility tht his or her score exceeds 84? 4.75 A soft-drink mchine cn e regulted so tht it dischrges n verge of μ ounces per cup. If the ounces of fill re normlly distriuted with stndrd devition.3 ounce, give the setting for μ so tht 8-ounce cups will overflow only % of the time The mchine descried in Exercise 4.75 hs stndrd devition σ tht cn e fixed t certin levels y crefully djusting the mchine. Wht is the lrgest vlue of σ tht will llow the ctul mount dispensed to fll within ounce of the men with proility t lest.95? 4.77 The SAT nd ACT college entrnce exms re tken y thousnds of students ech yer. The mthemtics portions of ech of these exms produce scores tht re pproximtely normlly distriuted. In recent yers, SAT mthemtics exm scores hve verged 48 with stndrd devition. The verge nd stndrd devition for ACT mthemtics scores re 8 nd 6, respectively. An engineering school sets 55 s the minimum SAT mth score for new students. Wht percentge of students will score elow 55 in typicl yer? Wht score should the engineering school set s comprle stndrd on the ACT mth test? 4.78 Show tht the mximum vlue of the norml density with prmeters μ nd σ is /(σ 2π) nd occurs when y = μ Show tht the norml density with prmeters μ nd σ hs inflection points t the vlues μ σ nd μ + σ. (Recll tht n inflection point is point where the curve chnges direction from concve up to concve down, or vice vers, nd occurs when the second derivtive chnges sign. Such chnge in sign my occur when the second derivtive equls zero.) 4.8 Assume tht Y is normlly distriuted with men μ nd stndrd devition σ. After oserving vlue of Y, mthemticin constructs rectngle with length L = Y nd width W = 3 Y. Let A denote the re of the resulting rectngle. Wht is E(A)?

29 4.6 The Gmm Proility Distriution The Gmm Proility Distriution Some rndom vriles re lwys nonnegtive nd for vrious resons yield distriutions of dt tht re skewed (nonsymmetric) to the right. Tht is, most of the re under the density function is locted ner the origin, nd the density function drops grdully s y increses. A skewed proility density function is shown in Figure 4.5. The lengths of time etween mlfunctions for ircrft engines possess skewed frequency distriution, s do the lengths of time etween rrivls t supermrket checkout queue (tht is, the line t the checkout counter). Similrly, the lengths of time to complete mintennce checkup for n utomoile or ircrft engine possess skewed frequency distriution. The popultions ssocited with these rndom vriles frequently possess density functions tht re dequtely modeled y gmm density function. DEFINITION 4.9 A rndom vrile Y is sid to hve gmm distriution with prmeters α>nd β>if nd only if the density function of Y is y α e y/β f (y) = β α Ɣ(α), y <,, elsewhere, where Ɣ(α) = y α e y dy. The quntity Ɣ(α) is known s the gmm function. Direct integrtion will verify tht Ɣ() =. Integrtion y prts will verify tht Ɣ(α) = (α )Ɣ(α ) for ny α> nd tht Ɣ(n) = (n )!, provided tht n is n integer. Grphs of gmm density functions for α =, 2, nd 4 nd β = re given in Figure 4.6. Notice in Figure 4.6 tht the shpe of the gmm density differs for the different vlues of α. For this reson, α is sometimes clled the shpe prmeter ssocited with gmm distriution. The prmeter β is generlly clled the scle prmeter ecuse multiplying gmm-distriuted rndom vrile y positive constnt (nd therey chnging the scle on which the mesurement is mde) produces FIGURE 4.5 A skewed proility density function f ( y ) y