Nevertheless, there are well defined (and potentially useful) distributions for which σ 2

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1 M. Meseron-Gibbons: Bioalulus, Leure, Page. The variane. More on improper inegrals In general, knowing only he mean of a isribuion is no as useful as also knowing wheher he isribuion is lumpe near he mean or sprea more evenly aross he sample spae; in oher wors, wheher he p..f. is sharply peake or relaively fla. For ha, we nee an inex of ispersion. In his leure, we efine one. Aoringly, onsier he ispersion ensiy D efine by D(x) (x µ) f(x), (.) where f is he p..f. of a ranom variable X on [, ) an µ is is mean. The funion D is never negaive an has he propery ha is value is small eiher if x is very lose o µ or if f is very small. Only if here is a signifian probabiliy of X being far from he mean an Area(D, [, )) be large. So a suiable inex of ispersion is In(D, [, )), alle he variane of he isribuion an enoe by σ. Tha is, he variane is σ (x µ) f(x) x. (.) The noaion iniaes ha variane is measure in square unis; e.g., if µ is in mm, hen σ is in mm. The square roo of he variane, σ, is alle he sanar eviaion. I has he avanage of being in he same unis as he mean. Inee for many purposes a beer inex of ispersion han eiher σ or σ is he oeffiien of variaion κ σ µ, (.) whih has he avanage of being a imensionless raio. Variane is illusrae by Figure, where D is graphe nex o he orresponing p..f. for Weibull isribuions wih shape parameers, an, respeively. A eah level, he unshae area in he lef-han panel is, whereas he shae area in he righ-han panel i.e., he variane inreases as he p..f. flaens ou. Alhough () efines he variane, i is rarely use o alulae i (unless, as we will isuss in Leure 8, he isribuion is symmeri), beause (.) implies (x µ) f(x) x (x xµ µ )f(x) x x f(x) x µ x f(x) x µ f(x) x (.4) Hene, from () an In(f, [, )), x f(x) x µ µ µ. σ x f(x) x µ. (.) Variane is a well efine measure of ispersion for all ommonly use isribuions. Neverheless, here are well efine (an poenially useful) isribuions for whih σ is no a finie quaniy, even if µ is well efine.

2 M. Meseron-Gibbons: Bioalulus, Leure, Page To illusrae his poin, an a he same ime show how a variane alulaion ypially invokes he funamenal heorem of alulus, we onsier he isribuion efine on [, ) by or F() f() F () θ A(θ ) (θ ) A θ A(θ ) (θ ) A (θ )(θ ) A(θ ) θ (θ ) A θ( A) (θ )(θ ) θ if if < {A(θ ) θ} if (θ ) θ if < (.) (.) wih θ θ < A < (.8) o ensure ha f is a p..f. (see Exerise ). In Figure his isribuion is fie o he prairie-og lifespan aa from Table 9. wih θ.98,.8 an A.4 (so ha θ/(θ).49, an (8) is saisfie). Minimum oal error,.4, is now more han wie as large as in Figure 9. an almos hree imes as large as in Figure 4.. Neverheless, he fi isn' wholly unreasonable. I a leas suggess ha he isribuion is a poenially useful one, whih is all ha Figure aims o ahieve. Before we an alulae he variane, we mus firs of all alulae he mean. On using (), we have {θ A(θ )} A {A(θ ) θ} if (θ ) f() (θ ) θ θ( A) if < (θ )(θ ) So, from (.) an (.) in onjunion wih Table 8., µ f() f() f() (.9) {θ A(θ )} A {A(θ ) θ} θ( A) θ (θ ) (θ ) θ (θ )(θ ). (.) The firs inegral in () is quie sraighforwar: by he funamenal heorem (in Leibniz noaion), i reues o {θ A(θ )} A {A(θ ) θ}4 4(θ ) 8(θ ) {θ A(θ )} 4(θ ) A {A (4 A)θ (9 A)θ } 4(θ )(θ ). {A(θ ) θ}4 8(θ ) (.)

3 M. Meseron-Gibbons: Bioalulus, Leure, Page The seon inegral, however, requires some are. In his regar, bu also wih a view o obaining a more general resul for improper inegrals, we efine u on [, ) by u() α. (.) Then, wih α θ, In(u, [, )) is he quaniy o be evaluae in (). For.8, u is graphe in Figure on suessively larger subomains [, K], for α in he lef-han olumn, an for α / in he righ-han olumn. In he eah ase, he shae area is In(u, [, K]). In he lef-han olumn, u ereases rapily enough ha, as K inreases, oal shae area remains finie. In he righ-han olumn, on he oher han, u ereases so slowly ha, as K inreases, oal shae area keeps on growing. In he firs ase, we say ha he inegral onverges, an ha he area is finie. In he seon ase, we say ha he inegral iverges, or ha In(u, [, )). We suspe ha here is a riial value of α, beween / an, a whih he shae area eases o grow wihou boun an insea onverges. To onfirm his suspiion, noe ha, by he funamenal heorem an (.), K K K α (α ) for any finie K. Now allow K o beome infiniely large. If α >, hen K (α ) beomes infiniely large as well, so ha / K (α ) approahes zero an () reues o u() (α ) (α ) K (α ) (α ) α α α α (α ) (.) α (α )K α (α ) in he limi as Κ. If α <, however, hen () beomes K α. (.4) K K ( α) ( α), (.) whih grows wihou boun. Wha happens if α is preisely equal o? Then α (α ) α K α K K ln(); an beause he ln(k) infiniely large as K approahes infiniy. The upsho is ha α α > an hene, on seing α θ, ha µ is finie only if θ >. Then, from (), () an () wih α θ, we have µ {A (4 A)θ (9 A)θ } θ( A)θ 4(θ )(θ ) (θ )(θ ) (θ ) {9θ A(θ )} θ 4(θ ) afer simplifiaion. For example, aoring o he moel of Figure, mean prairie-og lifespan is.449 years. Having alulae he mean, we now proee o alulae he variane. From () or (9), we have ln() K ln() ln(k) (.) logarihm is a srily inreasing funion, beomes if α (α ) if α (.), (.8)

4 M. Meseron-Gibbons: Bioalulus, Leure, Page 4 f() Thus, on using (), {θ A(θ )} (θ ) A θ( A) (θ )(θ ) µ σ f() f() f() {A(θ ) θ}4 if (θ ) θ if < (.9) {θ A(θ )} A {A(θ ) θ}4 (θ ) (θ ) θ( A) θ (θ )(θ ) (θ ). (.) On seing α θ in (), we fin ha σ is finie only if θ >, or θ >. Then, from (), () an he funamenal heorem, µ σ {θ A(θ )} A4 {A(θ ) θ} (θ ) (θ ) {θ A(θ )} (θ ) A4 {A(θ ) θ} (θ ) θ( A)θ (θ )(θ ) θ( A) (θ )(θ )(θ ) (θ ) θ {A 4(9 A)θ ( A)θ } θ( A) (θ )(θ ) (θ )(θ )(θ ) {θ A(θ )}, (.) (θ ) afer simplifiaion. Now, from (8), we obain σ {θ A(θ )} (θ ) {9θ A(θ )} (θ ). (.) Noe in pariular ha if < θ, as in Figure, hen he mean exiss, bu no he variane. As we remarke above, however, mean an variane boh exis for all of he isribuions we ommonly use. One example is he Weibull isribuion. Anoher example is he Gamma isribuion. The p..f. of he Gamma wih shape parameer an sale parameer s was efine in Exerise.9 by f(x) x e x/s s Γ(). (.) The Gamma isribuion has numerous biologial appliaions; for example, Troy an Robson (99, p. 4) use i o moel variaion among inerspike inervals (imes beween aion poenials) for mainaine isharges of a reinal ganglion ells. General expressions for is mean an variane are herefore of ineres. From () an (.), he mean is x e x/s µ xf(x)x s Γ() x x e x/s x s Γ(). (.4) Beause () efines a isribuion for any ( ) an s (> ), however, we mus always have In(f, [, )). Hene

5 M. Meseron-Gibbons: Bioalulus, Leure, Page for any or s, implying x e x/s s Γ() x s Γ() x e x/s x (.) x e x/s x Bu if () is rue for, hen i mus also be rue for. Therefore x e x/s x s Γ( ). (.) s Γ(). (.) Subsiuing in (4), we fin ha µ s Γ( ) s Γ() s, (.8) on using he Gamma funion's reursive propery Γ(r ) rγ(r) (.9) (wih r ). Similarly, beause () mus hol for if i hols for, Thus x f(x)x x e x/s x s Γ( ). (.) x e x/s s Γ() x s Γ() x e x/s x s Γ( ) s Γ(). (.) Bu seing r in (9) yiels Γ( ) ( )Γ( ) ( )Γ(), beause Γ( ) Γ(). So () implies x f(x)x s ( ). (.) Now (), (8) an () imply σ s ( ) s s. (.) Similarly, o alulae he variane of a Weibull isribuion, for whih we firs obain f(x) s (x / s) e (x/s), (.4) x f(x)x x(x / s) e (x/s) x s u e u u s x e x x, (.) afer using he subsiuion u φ(x) x/s as in Leure ; see Exerise. Subsiuing u x furher reues () o x f(x)x s u / e u u s u (/) e u u s Γ( / ), (.) again as in Leure, an again on using he efiniion of Γ; see Exerise. Moreover, µ sγ( /) from (.4). So () an () imply

6 M. Meseron-Gibbons: Bioalulus, Leure, Page σ s Γ( / ) s {Γ( / )}. (.) For example, if hen, from Figure.4, we have Γ( /) Γ(.).9 an Γ( /) Γ(.).98. So () implies σ.98 s.9s. s. Tha is, he op shae area in Figure is. s. Similarly, σ. s if beause Γ(4/).8998 an Γ(/).9; an σ.4 s if beause Γ(.).88 an Γ(), so ha he mile an boom shae areas in Figure are, respeively,. s an.4 s. Noe ha he variane inreases as he shape parameer ereases. Collaing our resuls, we fin from (8) an () ha µ s, σ s (.8) for he Gamma isribuion; whereas, from (.4) an (), µ sγ( / ), σ s Γ( / ) {Γ( / )} (.9) for he Weibull. In boh ases, he oeffiien of variaion epens only on he shape parameer: from () an (8)-(9), for he Gamma isribuion bu κ (.4) Γ( / ) κ (.4) {Γ( / )} for he Weibull. In eiher ase, he oeffiien of variaion is a ereasing funion of he shape parameer; e.g., (4) yiels κ. for, κ.4 for an κ. for (Figure ). In Figure 4, κ is ploe agains for boh isribuions. Referene Troy, J.B. & J.G.Robson (99). Seay isharges of X an Y reinal ganglion ells of a uner phoopi illuminane. Visual Neurosiene 9, -.

7 M. Meseron-Gibbons: Bioalulus, Leure, Page Exerises. Verify ha () efines a p..f. if (8) is saisfie. Hin: Esablish ha Min(f, [, ]).. Show ha boh mean an variane of he isribuion efine by () are posiive if hey exis.. Verify ()-()..4 A probabiliy ensiy funion is efine on [, ) by f() A( ) θ( A ) (θ ) θ( A θ ) (θ ) if if < where A, an θ are posiive numbers saisfying A <. (i) Does his isribuion have a finie mean? If so, wha is i? (ii) Does his isribuion have a finie variane? If so, wha is i?. The probabiliy ensiy funion of a isribuion on [, ) is f efine by x if x < f(x) ( x) if x < if x < (i) Fin he mean, µ (ii) Fin he variane, σ (iii) Deue ha he oeffiien of variaion is κ Fin boh he variane an oeffiien of variaion of he isribuion efine in Exerise... Fin boh he variane an oeffiien of variaion of he isribuion efine in Exerise..

8 M. Meseron-Gibbons: Bioalulus, Leure, Page 8.8 The p..f. of a isribuion on [, ) is f efine by Fin f(x) x ( x) if x < if x < if x < (i) he mean, µ (ii) he variane, σ (iii) he meian, M, an (iv) he moe, m. (v) Show ha he oeffiien of variaion is κ..9 The p..f. of a isribuion on [, ) is f efine by Fin f(x) x x if x < if x < if x < (i) he mean, µ (ii) he variane, σ (iii) he meian, M, an (iv) he moe, m. (v) Show ha he oeffiien of variaion is κ Fin boh he variane an oeffiien of variaion of he runae exponenial isribuion efine in Exerise..

9 M. Meseron-Gibbons: Bioalulus, Leure, Page 9.* The p..f. of a isribuion on [, ) is efine by / if x < x if x < if x < an L is a onsan. f(x) L g(x) where x g(x) (i) Fin L (ii) Fin µ, he mean (iii) Fin σ, he variane (iv) Show ha he oeffiien of variaion is κ (v) Fin F, he umulaive isribuion funion, on [, ] (vi) Show ha he meian is M. The p..f. of a isribuion on [, ) is efine by x)/ L if x < ( x)/ L if x < if x < where L is a onsan. Fin The oeffiien of variaion The mean The umulaive isribuion funion The variane The meian x(4 f(x) (i) L (iv) (ii) (v) (iii) (vi). A smooh probabiliy ensiy funion f is efine on [, ) by f() A. B C if < if 4 if 4 < (i) Fin he values of A, B an C (ii) Fin he mean of he isribuion (iii) Fin he meian of he isribuion, a leas approximaely (iv) Fin he umulaive isribuion funion (v) Fin he variane (vi) Fin he oeffiien of variaion

10 M. Meseron-Gibbons: Bioalulus, Leure, Page.4 The p..f. of a size isribuion on [, ) is efine by f(x) L g(x) where x if x < g(x) ( x) if x < 4 if 4 x < an L is a onsan. if x < (i) Wha mus be he value of L? (ii) Fin M, he meian (iii) Fin µ, he mean (iv) Fin σ, he variane (v) Show ha he oeffiien of variaion is κ /. (vi) Fin F, he umulaive isribuion funion, on [, ) (vii) Wha is he probabiliy of a size beween an?. The p..f. of a isribuion on [, ) is efine by f(x) L g(x) where g(x) x 4 x 4 4 x if x < if x < if x < 4 an L is a onsan. if 4 x < (i) Wha mus be he value of L? (ii) Fin M, he meian (iii) Fin µ, he mean (iv) Fin σ, he variane ( ) / (v) Show ha he oeffiien of variaion is κ / (vi) Fin F, he umulaive isribuion funion, on [, ) (vii) Wha is he probabiliy of a size beween an?

11 M. Meseron-Gibbons: Bioalulus, Leure, Page Answers an Hins for Selee Exerises. The mean exiss if θ >, in whih ase, (8) has he sign of 9θ A(θ). Beause A <, by (8), his quaniy mus exee 9θ (θ) (θ ), whih is posiive. Similarly for he variane..4 (i) µ f() f() f() A( ) θ( A ) (θ ) θ( A ) θ θ (θ ) So, seing α θ in (), he mean is finie only if θ >. Then µ A( ) θ( A ) (θ ) θ( A ) θ (θ ) A ( ) θ( A ) A θ( A ) (θ ) (θ ) θ( A ) (θ )(θ ) (ii) µ σ f() f() f() θ( A ) θ (θ ) θ (θ ) θ θ A (θ ) (θ ) A( ) θ( A ) (θ ) θ( A ) θ (θ ) (θ ) So, seing α θ in (), he variane is finie only if θ >, or θ >. Then µ σ A( ) 4 θ( A ) (θ ) θ( A ) θ (θ ) A 4 ( ) θ( A ) (θ ). (θ ) θ θ( A ) (θ )(θ ) A4 θ( A ) (θ ) A4 θ( A ) (θ ) afer simplifiaion, implying {θ (θ )A } σ (θ ), θ( A ) (θ )(θ ) {θ (θ )A }. (θ )

12 M. Meseron-Gibbons: Bioalulus, Leure, Page. (i) Define g on [, ) by g(x) Then f(x) g(x)/, an so x x if x < if x < if x < Bu µ xf(x)x xf(x)x xf(x)x xf(x)x xg(x)x xg(x)x x x (x x )x. So µ 4/. x x { x }x x { } x x { x x }x ( ) ( ) (iii) Similarly, x g(x)x x x (x x )x So x4 x { x4 }x { } x 4 x4 { x x 4 x4 }x 8 ( 4 ) ( 4) 8 4. σ x f(x)x µ g(x) x g(x)x σ µ x x σ (iv) σ 8 κ σ/µ σ 4, κ

13 M. Meseron-Gibbons: Bioalulus, Leure, Page. σ 4, κ.8 (i) Define g on [, ] by g(x) 4x x if x < if x < if x < Then f(x) g(x)/, implying µ xf(x)x xg(x)x 4x x (x x )x xg(x)x. Bu 4 x x { 4 x }x x { } x x 4. So µ /. (ii) Similarly, { x x }x x g(x)x 4x x (x x )x So σ (iii) Beause x 4 x {x4 }x x { } x 4 x4 { x 4 x4 }x. x g(x)x µ 4 4 In(f,[,)) Area of riangle of heigh f() wih base 4 4 exees /, implying M >, o fin he meian we solve / In(f, [M, )) In(f, [M, ]) ( M)f(M)/ ( M) /. So M, or M.84. (iv) From he riangular shape of he graph of f, i is lear ha m. (v) σ κ σ/µ.4.

14 M. Meseron-Gibbons: Bioalulus, Leure, Page 4.9 (i) Define g on [, ] by g(x) x x if x < if x < if x < Then f(x) g(x)/, implying µ xf(x)x xg(x)x x x (x x )x xg(x)x. Bu x { x }x x x x x { } So µ 4/ 8/. (ii) Similarly, x g(x)x { x x }x x x (x x )x x { x4 }x { x x 4 x4 }x x4 { } x 4 x4 So σ (iii) Beause x g(x)x µ In(f,[,)) Area of riangle of heigh f() wih base exees /, implying M <, o fin he meian we solve / In(f, [, M]) Mf(M)/. So Mf(M) or M /, implying M /.4. (iv) From he riangular shape of he graph of f, i is lear ha m. (v) σ κ σ/µ Go o hp:// (Problem #)

15 M. Meseron-Gibbons: Bioalulus, Leure, Page. (i) Define g on [, ] by x) ( x) Then f(x) g(x)/l, an In(f, [, ]) x(4 g(x), so L g(x)x if x < if x < if x < g(x)x {4x x }x ( x)x x {x x }x x { ( x) }x ( ) { ( ) } x x { ( x) } 8 4 (ii) µ xf(x)x x g(x) L x Bu xg(x)x {4x x }x (x x )x x { 4 x 4 x4 }x L xg(x)x x {x x }x 4 xg(x)x. 4 x 4 x4 {x x } { } So µ (iii) Similarly, x g(x)x {4x x 4 }x (x x )x x {x4 x }x x 4 x {x 4 x4 } x {x 4 x4 }x { 4 4 } So

16 M. Meseron-Gibbons: Bioalulus, Leure, Page σ x f(x)x µ g(x) σ µ x L x L x g(x)x σ 98 9 (iv) σ (v) Suppose. Then F() f(x)x κ σ/µ /. 4 L g(x)x 4 {4x x }x x {x x }x 4 x x ( 4 ) 4 ( ) Noe in pariular ha F() /. Thus, for, ( ) F() f(x)x f(x)x f(x)x F() 4 ( x)x 4 { ( x) } 4 L g(x)x x { ( x) }x 4 ( ) { ( ) } { } 8 ( ). (vi) F() / is less han /, so < M <. Thus F(M) / 8 ( ) ( M) 4 4 M..

17 M. Meseron-Gibbons: Bioalulus, Leure, Page.4 (i) L In(g, [,]) Area uner quarilaeral (ii) M /4 (iii) µ 9/ (iv) µ σ /4, implying σ /94 (v) σ 9/ 94 9/, implying κ σ/µ / (vi) From ( x) an F(x) In(f, [, x]) (x ) ( x) F() F() 4/ /4 / f(x) we have 4 x F(x) (vii) x if x < if x < 4 if 4 x < if x < if x < if x < 4 if 4 x < if x <. L In(g, [,4]) Area uner re-enran quarilaeral (ii) M 4/ (iii) µ / (iv) µ σ /, implying σ / (v) σ /, implying κ σ/µ / (vi) From (4 x) an F(x) In(f, [, x]) x (4 x) F() F() / / / f(x) we have x F(x) x (vii) 8 x 8 x 8 if x < if x < if x < 4 if 4 x < if x < if x < if x < 4 if 4 x <