PROPERTIES OF TWO NEWLY-DEFINED DISTRIBUTIONS AND THEIR RELATIONSHIPS TO OTHER DISTRIBUTIONS
Jiang-Ming Wu, University of Wisconsin - Madison
Abstract If random variable X has the following pdf then we say that it has thc sine distribu-
Two newly-defined disu'ibutions,sine distribution and cosine distribution, have been pro- tion and denote it by X - S( x,m ). posed in this article. Their basic properties including moments, cumulants, skewness, fix) = s(x;m) = -~-smm . (nu:) if 0 < x < ~"~'m. m>0 kunosis, mean deviation and all kinds of generating functions are discussed. Besides, = 0 otherwise. (2.D their relationships to other distributions are also presented including proofs. It is seen that cosine distribution can serve as a very rough approximation to the standard normal 2.2 Cosine distribution distribution under suitable condition of parameter chosen. If random variable Y has the following pdf then we say that it has the cosine distri-
bution and denote it by Y - Cos( y;m ). KEY WORDS :
Sine distribution;Cosine distribution;Transformations of random variables. g(Y)
1. Introduction = 0 otherwise. (2.2)
It is well known that the pictures of sine and cosine function look like the following It is no doubt that expressions ( 2.1 ) and ( 2.2 ) are two pdfs since one can easily figures. verify this just by integrating them over their corresponding ranges. Because the ui- Fisure i Figur'e 2 gonomeu'ic functions are periodic and satisfy
to to /(x) =ffx+2nn) n ~ Z, (2.3) we can expand those previous two delinifions as follows.
2.3 Displaced sine distribution
If random variable X has the following pdf then we say that it has the displaced sine
distribution and denote it by X - Sd( x;m.n ).
m . fix) = sd(x;m,n) = -~--sm(rex) if 0 < x _2nm < -'~m VV VV m>O,n~Z to to • . . , .... , 7 T = 0 otherwise. (Z4) -B -4 O 4 13 -(] -4 O 4 [3
x x 2.4 Displaced cosine distribution SINE FUNCTION COSINE FUNCTION If random variable Y has the following pdf then we say that it has the displaced If we integrate the shaded part of both figures, we'll have cosine distribution and denote it by Y - Cd( y;m.n ).
isi,t~= --CO,Lg t ~3 zl 2 (1.1) g(y) - cdfy;m.n) - ~co,
2 : 0 otherwise. (2.5)
-T 2 In this article we will discuss only those properties of the sine and cosine distribu- Therefore if we let fix) = ~smx1 . and g(x) = ~-cosI x , then both fix) and g(x) can be tions since the other two distributions can be obtained just by the linear transformation of treated as probability density functions ( abbrev, pdf hereafter ) under suitable ranges. these two distributions. Besides, all their propenies arc almost the same, thetetbre, it is Burrows (1986) discussed the exurcme statistics from g(x) which he called the sine unnecessary to have a discussion here. distribution. Expanding these basic ideas I have defined two new distributions with fix) 3. Basic properties and g(x) as their special cases. The main goal of this article is to introduce these new 3. l Relationships among moments distributions and their basic properties. Also. their relationships to other distributions will Let IX',.= E(X') and IXr = E (X-IX)". Then wc have the following two lemmas con- be discussed and will play a very crucial role in this article. A very rough normal ceming the moment's relationship within these new distributions. approximation has been proposed and its comparison to Chew's approximation ( Chew Lemma 3. I. I If X - S( x;m ) then all moments of X exist and satisfy the following rcl',- 1968 ) is also given. The names of these distributions are Sine distribution and Cosine tion : distribution due to their corresponding functional forms of pdfs. Finally, a diagram, due to Lcemis ( 1986 ), showing relationships to others for these two distributions is IX'r = l(~-)r - ~IX',.2 r=1,2,3 ..... (3.1.1) presented in the appendix. In equation (3. l.l) we have IX'-I = 0 and IX'0" 1.
pf:
2. Definitions I_ ml 2.1 Sine distribution IX..E(x').=- .'! x'si.(.~)~ let u reX" .dr m$in(m~tklx
356 x m By using formula (3.1.1) we can obtain It',.• = I. 2. 3, 4, for the sine distribution. Then by using formula (3.1.61 we can obtain 14,. and by using formula (3.1.75 we can
obtain ~, for r = 1, 2, 3.4. for the same distribution. Similarly. we can get p',. It,, & 4, . + :".~.(.=),~ ~ !'_L-~..-,..,(,=~,. for the cosine distribution. Their corresponding first four moments and cumulants are
exhibited in table 1 and table 2, respectively. 1(~)" r¢•-1~ mi = 2 m m a "~ x"asin (rex)¢L¢ From tables 1 and 2 we see that P2,143 and 144 of sine and cosine disu'ibution are the
same, as a consequence, their !c2, 43, and to,t values are also the same. = --,,._(._..)" _ •(r-t) 14, QecL 2 m m~ ,-z 3.3 Skewness. kurtosis and the mean deviation Lemm~ 3.1.2 If X - Cos( x;m ) then all moments of X exist and satisfy Since sine and cosine distribudons have the same values of It:. 143 and P t. they v, ill 14; = 0 if r Lr odd /.(•-|~) Table 1 The first four moments and cumulants of the sine distribution it', =( •~-~ ) ' -~ It' ,_ 2 ifr b even #I" ...... r 1 2 3 4 14'r it 112--4 lt'~-6t~ ~14-12~z+48 or l.Ca = (.~)ze - 21: (2k-l)m 2 14'~-2 k = I .2.3...... (3. 1.2) 2m 2rn 2 2an 3 2.m 4
n2-8 Z¢4-48n2+384 p, 0 ~ 0 pf: 4m 2 16m 4
.Z..x R n2-8 -n*+96 7an ~ 4m z ~m 4 m 14; =E(X')= T j' xrcos(~)ax let u =x r ,dr =cos(mx)dx I[ Table 2 The first four moments and cumulants of the cosine distribution
r [ 2 3 4 ' ~_m x-~--.sin• (ms) I ._ rxr-l sin (mr) '" nz,K ' n.~...48ttz,384 2m -'~ 14, 0 --r¢2-8 0 rc4-4glt2+384 4m 2 16m 4
=7 (-~5 +( 5" -T , 4, 0 z12-'---~8 0 -z14+96 2m ._ 4m 2 Rnt 4 ,
also have the same values of the coefficient of skewness o.3 and the kurtosis ct4. The == - -~.l---.---cos.~#tc )t " + m T (~5 +(---~)" ~'" " "' "~ corresponding definitions for a s and cq are
x..F_ 143 143 (Z3 == ~'1 = p'~ = "~", (3.3.11
2#, 14.._L 144 (3.3.2) When r - 2k-! is odd, the right-hand side of equation (3.1.3) is 0, so It'~-t = 0. ~,= ~,= 141 = ~-. Therefore. from table I and 2 we have When r - 2k is even, then equation (3.1.3) is reduced to equation (3.1.25 which is the desired result. eli =0. (3.3.3) Qed. z~'L-.48K2+384 = 2.19 (3.3.4) °c:s " tt 8-16rC2+6,4. Two other well known general results about moments and cumulants are obtained for both dismbutions. from Johnson & Kots (1969). They will be used to find the firstfour (centraD moments One can expect that their mean deviations must also be the same. If X - S( x;m ) and eumulants of these new dismbutions. then Lemma 3.1.3 Let X be any random variable with finite moments, then
v t =E[IX-E(X)II = E[IX-~- II (3.1.4) j.o x m or 14. = ~(j") ~'i ~-~ . (3.1.5)
From equadon (3.1.4) after plugging in r = 2.3,4 respectively, we obtain x__ =_. 2m m m ! (~-x 5sin (rex ~x + Tm !(x-~Sxin(m.l;)dx 142 = 14'=- 14'~, =T 2m I,q = It'3 - 314'z It't + 214"t 3 , 2m 2m p~ = rt'4 - 4 It'3 14't + 6 It'z 14'~ - 314'4. (3.1.6) Lemma 3.1.4 Let X be any random variable with finite moments, then
let =it't = E(X), 2m 21,1 Kz =Itz = Vat(X). = m ! _"-L-sinZm (rex )dr.- m !xsin(mx)d.~;
~3 "143,
~¢4 = It4 - 3 14 zz . (3.1.7) - --~cos (nix) I ~ 0 -m -Xcos(mx), 2~ + cos(~, o 3.2 The first four moments and cumulants
357 I ~¢-2 (3.3.5) 2m m 2m m2(e 2m + • ~ ) m 2 ! my(t) .... cos/+( z ) (3.4.4) 2(m 2 + t:) m 2 + t2 Similarly, if Y - Cos( y;m ) then ~gy(t ) = Iog[my(t )] = log mZ +'"ta cosh( ~--~-t <3.-~.5) v t =E[IY-E(Y)I] =E[IYI] =_. 211 m 2 1[ m (~y (t ) = my.(it ) = m 2 _ t2 gosh (i.~.Tt ) (3.4.6) = T S 'Y ~cos (my)@ &ITJ 2m 4. Relationships to other distributions 2#1 = m ! ycos (my)dy This section contains the most important topic of this article and its goal is to
demonstrate how these two distributions relate to others. For convenience, they will be
discussed separately and proofs of the relationships will be given only for the sine dis~i- =m ~mSin(my)12~ - ?'-~sin(my) o button case.
Q IC TC I R.-2 4. l Sine distributioncase =-- + ).Lcos (my) I ~ = .~.- ~. =-~-- (3.3.6) 2,m m All relationships ate based on the transformation of random variables. That is, if X
In addition, we get an extra result from equation (3.3.5) and (3.3.6), that the ratio of is a random variable with pdf fix) and Y is another random variable obtained from X by
their mean deviation to their standard deviation is also the same and is independent of the the transfom',adon Y - h(X) ,, AX, then pdf of Y is given by
parameter m, i.e., is a constant : 1 g(y) =[(A'ty)lJ l =f(A"ty) Ide[A I ' (4.l.l) vt ~-2 = ~ = 0.835. (3.3.7) where J is the Jacobian of the maasformation. "q~':-8 The tint three relationships axe fairly easy to see. so their proofs will be skipped 3.4 Characteristic and generating functions here. Lemma 3.4.1 If X - S( x;m ) then its moment generating function rex(t), cumulant gen- It R4.1.1 If X - S( x;m ) then Y - X - ~- - Cos(y;m). erating function ~.,¢ (t) and the characteristic function #x (t) arc
R4.l.2 if X - S( x;m ) then Y = X - 2n., Sd(y;m,n). In fact. the sine dismbution is a m mY(e "~ +1) m2 n (3.4.1) m x (t) = = .--~--=e "~' • cos/, (_=---~) 2(m: + t 2) m,~ + t, 2m speci',d case of the displaced sine dismbution, i.e. S( x;m ) a Sd( x:m,O ).
R4.1.3 If X - S( x;m ) then Y = -cos( mX) - U( -I.l ) which is unifon'nlv distributed m2(e ~ +l)] (3.4.2) Vx(~) = l°g[mx (~)l =~°~ 2i,,,~+,:) j over the interval [ - 1,1 ].
.x % (3.4.3) R4.1.4 If X - S( x;m ) then random variable Y = tan[-~-costm.X')l is distnbutcd as the
respectively. standard Cauchy distribution C( O, l ) with pdf
pf: We only need to prove that equation (3.4.1) holds. l '+¢")" ,,(i+.,,:'~ -<' <" (4.,.,> m m . pf : X - S( x;m ) then fix) - ..~.-sm(mx), 0 < x < --. Since mx(t)=E(¢lX)= T+! e=sin(mx)dx m
R /g r - tan {--~cos (mX)l "+ y "tan {--~cos (rex)! = - e=co$(mx)l r~ + te=cos(mx)d ~ m -~ ,ly = sec 2 [--~cos(,ra1~ )i-~--s,nml~ . (mx)~
1 i ~+ IJI = I ~--..I -- = .~-(e" +l) + 2icos( mx )~x dy 2 ~ mt~ sec [-.~cos(mx )l-~--sin(mx)
111 . I l t e = # 0' ) .-[ (x )IJ l - /..:-sin(rex) =T ("" *t)+T ""m-'~'('=)' ~' - ~"=~"("=)m sec2 [-.~coa~ (mx )l--~--sinm~ (m.x )
I I I
= g-re" +t) - --~ ! e=sin(mx )~ • .sec2 t-Teas ~ (mx)] ~ l+tan2[-2co$ (rex)i
I I I (e -~' t2 14-22 lt(l+y 2) Qcd. = ,~ +l)--~mx(t) R4.1.5 If X - S( x:m ) then random variable Y - tan[.tan'l~..cos(mX)] is dis~buted as
t 2 1 "--'+' the doubly u'uncatcd Cauchy distribution at ;L with pdf •"+ mx(t)(l + "~'~') = ~-(e" +l)
i = .... ;L < y < ~. (4.1.3) g (Y) - .,,..,,,~tan'~'Jr~'"z' _+mx(t)= m2(e" +1) Qed. 2(m: + t2) pf: y = tan[-tan-IZcos(mx)] ~ IJl = I ,~ydx I - I ~dv i "t For the cosine distribution case, we have very similar results. So. I'll just list the 1 ~ IJl= formulas for these functions without any proofs. sec 2{-tan'iZcos (rex)Ira tan'lZa'Ot (re.l;) m . l Lemma 3.4.2 If Y - Cos( y;m ) then g Ix' ) =f (X)IJl = ,"=-sin (m.~) sec2[_tan.i__o$)~.. (rex )Ira tan-I~.in )
358 = 1 1 R4.1.9 If X - S( x;m ) then random variable 2tan-I~ l+tan:l-tan'Z~'os(ntr )1 lZ I Y = sin [-~cos {nix )! (4.I. zo) 2tan-~X( l+y 2) Qed. is distributed as the arcsine distribution with pdf R4.1.6 If X - S( x;m ) then random variable 1 &(Y)= ,"-"r--"~. Y = et + 2~tanh-I[-cos (rex)1 (4.1.4) =~l-y" -I
Hence pf : y = ct + 213tanh't[-cos (nu)i m 2 1 Qed. tanh(2Z2~) =--cos(n=) g o/ ) = f (x ) lJ I = -~sin (m.x ) m..... esin (mx ) l~_y 2 R4.1.10 lfX - N( 0.1 ) then random variable -+I-cos2(mx) = l-tanh'(2Z~)=sech:(Z-2--~-;) it Y " "~ + Isin't[2sgn(X)'i(m IXI)] (4.1.12) IJI = la~l = l"c°sZ(mx) dy 2[],m sin(rex) is distributed as the sine diswibudon. Where m . . . l-cos2fmx) sO / )If (X)IJI = T$|n ~mx) 12~lsin(//u~ ) /(z)= i 2"~'~'~e .,Z 2 dt (4.1.13) = -'-~[ l-.eos2(mx )] - --~sech 2( 2~ ) qcd. i .,4 l':h~-: i rlt.(~.qr:l I from z~ro to z t.llldr,,r t'ht, R t, arwlarrl nq)t"l~;l [ ctlr~,'~: R4.1.7 [f X - S( x;m ) then random variable sgn(x)=+l ifx 20,
Y = sinh-' [tan(--~-cos(.tX))] (4.1.6) =-1 ifx <0. (4.1.14') is distributed as the hyperbolic secant distribution with pdf pf: Let Z = ~sin'Z[2sgn(X).tflXL I)! then Y = Z 4- . From result R4. I. I, we need m 2t"~
only to prove that Z is cosine distributed, then the result follows. gO')= Isechfy) --- / Y = sinh'~[tan(-2cos(mJ:))l z = -~sin-Z[7,sgn(x).l(IxI)! therefore, when x 2 0 we have -.e X = lcos[-2tan-I(ainh O/))i z = ml---sin-t[2/(x)l. 1_2_.._ coshy IJl=ldxl= . ~ l+sinhZ)'1 Also, dy m'%/ l-[-~tan-t(sinh2 o/ ))l I(x)= • l dt then m ~in(mx) #(y)-/Or)l/I m Tman("=)mlff4in (/l~[)2zech(7) = Is¢chO'). Qcd. ,~. ±- 21'~) ~ = 26(x)_~ d.z R4.1.8 If X -- $( x;m ) then random variable m ,¢i..(~(x))Z moo, (ms ) ~jJ = i-~tiT; = m .cos fret ) Y = ¢xp[sinh-I[tan(-'~'cos(mX))il (4. t.8) 2O(x) is distributed as the h',df.Cauchydistribution with pdf Hence m 2 m.:.c,os.(m:,,), = T c°s (m:) = cos (z ;m ). &O/)a ~(~+F2) y >0 (4.1.9) &(:) =/(x)lJI =#(x) 2~(x) pf : We can use two different ways to prove this result, one is to use the transformation Similady, since I( -x ) = -I( x ), we have, when x < 0 directly as those given in this ,~etion. The other way is to use the previous result R4.1.7 z = Isin-t[-~(-x)l = lsin't[2/(x)l, m via the hyperbolic s~ant distribution. Here I give the second method of the proof. and the resultfollows after the same transformationis applied. Qed. 1 v Let Z = sinh't[tan(-2:co$(mX)) j from resultR4.1.7 and equation ( 4.1.8 ) we know R4.1.11 If Y = sin'l[sgn (T)'I r--(~,.'~-)]+ ,'-~.~_ is distributed as the sine distribution v÷T l that Y = exp( Z ) and Z has a hyperbolic secant distribution. Henceforth S( y;m ), then T - tv is t distributed with v d.f.. ,.,.here /(:) Isech(z) --- < z <+-- 1 z ixfp,q ) = B (p,q ).!yP-t(l-y)q't dy (4. l.tS~ Now we have is the incomplete beta function. y =c~p(z) --~ z =1,10,) 1 I 1 v n ~-sin- [sgn(T).l ta (~',~)i then by result R4.1.I we have Z - -.+ ijl=ldZ i= I pf: Let Z- Y - -~'= ~y 7 v+T~ Cos( z;m ) is cosine distributed. 2 1 i 2 *=~ g~)=/(z)lJl =f(/n(y))tJl .... = ...__.__.._ Qed. ;t y +y-i y n(1 +y2) Hence 359 arcsine distribution with pdf ( 4. I. I I ). : = lsin-t [sgn(t)'t ,, (-~,¥ m L ~ J R4.2.11 IfX - N( 0,I ) then Y - Isin'i[2sgn(X)'l(IX l)J - Cos(y;m), where If. ) and m =1 .-t 1 v sgn(. ) arc defined as in equation ( 4.1.13 ) and ( 4.1.14 ), respectively. = ~-sin- -t ,, (¥.¥ if t<0. R4.2.12lfT-tv, theny=lsin "t [agn(T).l r' (-~,'~ -Cos(y;m). m I, ~ J L ~ J Since Finally, since cosine distribution is symmetric about its mean 0, we can use it as a t 1 very rough approximation to the standard normal distribution N(0,1) after suitably choos- ~,.+= ._L v-..t I .Iv 1 2 2 (4.1.16) ing the value of the parameter m. To let the standa,rd deviation be unity, we need a(T, ¥) we have r~-8 4m--.--T = I. (4.2.1) iI Hence ~) t ,, .t ~ I ~' 2(t_y)- dy ";~(¥'T _. ~ v. ,. o "(T,TJ m - --~ = 0.6836. (4.2.2) t_ ., Table 3 gives the table of standard normal and two kinds of cosine distributions so .t (':) :(t-H-~',): ~ 2,, m R ( [ V'l vet" V÷/" (V+/2) 2 that we can see how good the approximation is compared to the real standard normal dis- "TT" tribution and the cosine distribution defined b.v Chew (1968'). _ v*._.! 2 .~) 2 = ~-~-(l+ Table 3 table of the standard normal and cosine distributions ~a Lv (7.T) x + Normal Cosl Cos2 ~-0 .5000 .5000 .5000 ~cre~re 0.2 .5793 .5720 .5682 0,4 .6554 .6442 .6350 g(t)= f (z)lJ I 0.6 .7257 .7088 .6994 0.8 .7881 .7702 .7600 v*.i 1.0 .84 i 3 .8252 .8158 2 (I+...~.)--T" !.2 .8849 .8728 .8657 5,, 1.4 .9192 .9122 .9088 m 1.6 .9452 .9436 .9422 = --cos (mz) , ~a(T, T ) 2 1.8 .9641 .9670 .9714 m l-I/ v_2~=(¥.¥)1 1 v 2.0 .9772 .9832 .9897 2.2 .9861 .9931 .9989 2.4 .9918 .9982 1.0000 2.6 .9953 .9998 2.8 .9974 1.0000 _w__L 3.0 .2 .9987 = "~(I+.~-~) 2 = iv(t) Qed. "x is multiple of the standard deviation :va 'v y (T.-i-) The Cosl distribution in the table is given by Chew (1968) with pdf l+cos x 4.2 Cosine distribution case f(z)= ~ 2== -~ax ~K (4.2.3) while the Cos2 distribution is defined in this article by m = 0.6836. i.e., with IxIf Since we have result R&I.I, the proof of those similar results will not be given here. The interested reader can prove all of them by using those transformations directly. /(x) -,0.3418 cos(O.6836x) -2.2978 tion. 98.92% is covered under a standard normal curve. Besides. when a value exceeds 2.3 m standard deviations then my approximation will no longer be valid. This is the reason pf: Since cos( x;m ) = .~cos(mx) is an even function, the result follows. why [ say that it is a very rough approximation. R4.Z2 If X - Cos( x;m ) then Y - X 4. ~- - S(y;m). APPENDIX R4.2.3 If X - Cos( x;m ) then Y' - X + 2n - Cd(y;m,n). In fact, we have Cos( x;m ) • nl Cd( x:m.0 ). ,.__.,.÷..,,, -= t R4.2.4 IfX - Cos( x.m ) then Y = sinfmX) - U(-I, I). £,.,,. +i+~,'~i R4.2.5 IfX - Cos( x:m ) then Y = tan[~-sm(mX)]. - C(0, 1). 't -.~,...t-.-+,t~ Ul R4.2.6 If X - Cos( x:m ) then random variable Y = tan [tan-lksin (rmY)] is distributed as the doubly truncated Couchy distribution at X with pdf ( .I.. 1.3 ). R4.2.7 If X - Cos( x:m ) then random variabl,: Y = ot + 2~tanh-l[.s'in (mX)l is distributed as the logistic distribution with pdf ( 4.1.5 ). R4.2.8 If X - Cos( x;m ) then random variable Y = sinh't[tan (2sin (reX))1 is distributed as the hyperbolic secant distribution with lxlf ( 4.1.7 ). R4.2.9 If X - Cos( x;m ) then random variable Y = exp [sinh't[tan(2sin (nO¢))ll is dis- tributed as the half-Cauchy distribution with pdf ( 4.1.9 ). R4.2. I0 If X - Cos( x;m ) then t-andom variable Y = sin [--~sin(rex)I is disu'ibuted as the I - .....::.-° I 360 or How to Relate Distributions? x =, (1/re)sin "1 ((2/to)sin "1 y). (8) Therefore, the relationship between random variable X and Y is Theorem Suppose random variable X has cdf F(x) and random variable Y has Y - sin((=/2)sin(mX)). (9) cdf G(y). Suppose further that F(.) and G(.) are continuous with inverse ps: see result R4.2.10. functions exist. If we equate F(x) and G(y) and solve for y in terms of x, Let X - S(x;m) then its cdf is we'll have y = G'I(F(x)). Then the transformation defined by Y = h(X) - G'I(F(X)) (1) F(x) - .5 -.Scos(mx) 0 < x < =/m. (10) or the inverse transformation Let Y be logistic distributed with cdf X - h "1 (Y) - F "1 (G(Y)) (2) G(y) - .5 +.5tanh((y-(z)/21~) -=o < y <=o. (11) is the relationship between these two random variables. Equating F(x) and G(y) we have Ex.1 Let X -- C(0,1) then we have .5 - .5cos(rex) = .5 + .5tanh((y-a)/213) F(x) = .5 + (1/~)tan'Ix -~< x Let Y -- U(-~/2,~/2) then y = o~ + 2~tanh'~(-cos(mx)) (12) G(y) = .5 + y/~ -=/2 < y < ~/2. (4) or Equating F(x) and G(y), we have x = (lira)cos "1 (-tanh((y-a)/2~)). (13) .5 + (lht)tan'Ix - .5 + y/~ Therefore, the relationship between random variable X and Y is or y - tan'ix. Y = o~ + 2~tanh'l(-cos(mX)). (4.1.4) Therefore, relationship between the uniform and the standard Cauchy ps: see result R4.1~6. distribution is Y - tan~X or X = tanY where X and Y have tile standard Cauchy distribution C(0,1) and the uniforn distribution over the interval REFERENCE [-x/2,Jt/2], U(-s/2,=/2), respectively. Burrows, Peter M. (1986), "Exurcrr~ Statistics From the Sine Distribution," The Ameri- Ex.2 Let X ~ Cos(x;m) then its cdf is can Statistician. 40, 216-217. F(x) -.5 +.5sin(mx) -~/2m < x < ~/2m. (5) Chew, V. (1968), "Some Useful Altemadves to the Normal Distribution," The American Let Y be arcsine distributed with cdf Statistician, 22, 22-24. G(y) = .5 +(1/,'t)sin'ly -1 < y < 1. (6) /ohnson, Norman L, and Kot,z, Samuel (1969), Discrete Distributions, Ncw York: lohn Then by letting Wiley. .5 + .5sin(mx) = .5 + (l/~)sin'ly Leemis, Lawrence M., (1986), "Relationships Among Common Univariate Distribu- we have y = sin((:c/2)sin(mx)) (7) tions," The American Statisdciaa, 40, 143-146. 361