Journal of Modern Applied Statistical Methods May, 2007, Vol. 6, No. 1, /07/$ On the Product of Maxwell and Rice Random Variables

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1 Journal of Modrn Applid Statistical Mthods Copyright 7 JMASM, Inc. May, 7, Vol. 6, No., /7/$95. On th Product of Mawll and Ric Random Variabls Mohammad Shail Miami Dad Collg B. M. Golam Kibria Florida Intrnational Univrsity Th distributions of th product of indpndnt random variabls aris in many applid problms. Ths hav bn tnsivly studid by many rsarchrs. In this papr, th act distributions of th product XY hav bn drivd whn X and Y ar Mawll and Ric random variabls rspctivly, and ar distributd indpndntly of ach othr. Th associatd cdfs, pdfs, and th momnts hav bn givn. Ky words: Mawlll distribution, products, Ric distribution. Introduction Th distributions of th product X Y, whn X and Y ar indpndnt random variabls, aris in many applid problms of biology, conomics, nginring, gntics, hydrology, mdicin, numbr thory, ordr statistics, physics, psychology, tc, (s, for ampl, Cigizoglu & Bayazit (), Galambos & Simonlli (5), Grubl (968), Ladarl, t al. (997), and Roach & Klijunas (97), among othrs, and rfrncs thrin). Th distributions of th product X Y, whn X and Y ar indpndnt random variabls and com from th sam family, hav bn tnsivly studid by many rsarchrs, (s, for ampl, Bhargava & Khatri (98), Mali & Trudl (986), Rathi & Rohrr (987), Springr & Thompson (97), Stuart (96), and Wallgrn (98), among othrs,). In rcnt yars, thr has bn a grat intrst in th study of th abov M. Shail is in th Dpartmnt of Mathmatics. mshail@mdc.du. B. M. Golam Kibria is an Assistant Profssor of Statistics at Florida Intrnational Univrsity. E- mail: ibriag@fiu.du. Th authors ar gratful to th Fu Jn Catholic Univrsity, Profssor W. L. Parn, National Chiao-Tung Univrsity, and National Scinc Council of Taipi for providing financial support and rsarch facilitis ind whn X and Y blong to diffrnt familis, (s, for ampl, Nadarajah (5), and Nadarajah & Kotz (5), among othrs). In this papr, th distributions of th product X Y, whn X and Y ar indpndnt random variabls having Mawll and Ric distributions rspctivly, hav bn invstigatd. Th drivation of th cdf, pdf, and th momnt of Z X Y involv som spcial functions, which ar dfind as follows, (s, for ampl, Abramowitz & Stgun, 97, Gradshtyn & Ryzhi,, and Prudniov, t al., 986, among othrs, for dtails). Th sris ( α α α β β β ) α α ( αp) z ( β) ( β) ( βq)! F,,, ;,,, ;z p q p q is calld a gnralizd hyprgomtric sris of ordr ( p, q), whr (α ) and (β ) rprsnt Pochhammr symbols. For p and q, w hav gnralizd hyprgomtric function F of ordr (, ), givn by F ( ; β, β ; z) ( α ) ( β ) ( β ) z α.! Th intgral,

2 ON THE PRODUCT OF MAXWELL AND RICE RANDOM VARIABLES α t t dt Γ α, α >, is dfind as a (complt) gamma function, whras th intgrals and γ α ( α, ) t t dt, α >, α α, ) t t dt, α >, whr Γ (.) dnots gamma function. Also, (, I ) + ) F +,+ ; whr F dnots th conflunt hyprgomtric function. Whn, modifid Bssl function of first ind, I, of ordr is obtaind as follows: 4 I () (!) ar rspctivly nown as incomplt gamma and complmntary incomplt gamma functions. For ngativ valus, gamma function can b dfind as n n ( ) Γ n (n ), whr n is an intgr (.g., Andrws, t al., 999, and Bohr & Mollrup, 9).Th rror function is dfind by For or and R + >, arg z < R ( z ), ; rf u du, th Bssl function was modifid of scond ind of ordr, givn by whras th complmntary rror, rfc dfind as rfc( ) u du rf., is Th modifid Bssl function of first ind, (), for a ral numbr, is dfind by I I 4 (!) + + ), z Γ K Γ + For arg on would hav z K For non-intgr, zt ( t ( z), R( z ) ) <, z t 4t dt. + ( t) dt.

3 SHAKIL & KIBRIA K { I I } sin( ) Th following Lmmas will also b ndd in our calculations. LEMMA (Gradshtyn & Ryzhi (), Equation (3.38.4), Pag 37). For R ( μ ) >, and R ( ) >, t μ t dt μ. ). LEMMA (Prudniov t al. (986), Volum, Equation (.8.5.5), Pag 6). For a >, α p / t t rf(ct)dt ( α+ )/ cp α+ 3 3+α Γ F ;, ;c p α+ α α α Γ F ;, ;c p α αc LEMMA 3 (Prudniov t al., 986, Volum, Equations..3.4, Pag 5). For R ( α) <, R( p ) >, R >, and R () c >, α p / γ(,c) d α + c (p) α) F ; +, +α+;cp Γα+ F ( α; α, α;cp α ) αc whr F dnots gnralizd hyprgomtric function of ordr (, ), (s dfinition abov). LEMMA 4 (Gradshtyn & Ryzhi (), Equation (3.47.9), Pag 34). For R β >, R γ >, β γ β d γ K ( β γ ) whr K (.) dnots modifid Bssl function of th scond ind, (s dfinition abov). Distribution of th Product Xy Lt X and Y b Mawll and Ric random variabls rspctivly, distributd indpndntly of ach othr and dfind as follows. Mawll Distribution: A continuous random variabl X is said to hav a Mawll distribution if its pdf f X () and cdf F X () ar, rspctivly, givn by f X 3 a ( y) a and 3 γ, a FX rf a whr γ ( a, ) and a, >, a > a (), (3) rf dnot incomplt gamma and rror functions rspctivly, (s dfinition abov). Ric Distribution: A continuous random variabl Y is said to hav a Ric distribution if its pdf f Y (y) is givn by y ( y + v )/σ yv fy( y) I, σ σ whr ( y) y>, σ >, v (4) I dnots th modifid Bssl function of th first ind, (s dfinition abov).

4 ON THE PRODUCT OF MAXWELL AND RICE RANDOM VARIABLES For v, th prssion (4) rducs to a Rayligh distribution. In what follows, w considr th drivation of th distribution of th product X Y, whn X and Y ar Mawll and Ric random variabls rspctivly, distributd indpndntly of ach othr and dfind as abov. An plicit prssion for th cdf of X Y in trms of hyprgomtric function has bn drivd in Thorm. In Thorm, anothr plicit prssion for th cdf of X Y in trms of hyprgomtric function and modifid Bssl function of th scond ind K () has bn drivd. Thorm Suppos X is a Mawll random variabl with pdf f X () as givn in () and cdf F X P( X ) givn by (3) in trms of th incomplt gamma function. Also, suppos Y is a Ric random variabl with pdf f Y (y) givn by (4) in trms of th modifid Bssl function of th first ind I ( y). Thn th cdf of Z X Y can b prssd as Fz /σ 4 σ σ (! ) 3 3 Γ a σ z az F ;, ; 3 4σ a + z (+ ) az F ;, ; + Γ (+ ) 4σ (4) for pdf of Ric random variabl Y, th cdf F z Pr X Y z can b prssd as Fz z Pr X Y z FX Y y f (y)dy / σ y / σ 3 az vy y, I γ dy σ y σ (5) whr y >, z >, a >, σ >, v. Th proof of Thorm asily follows by using dfinition () of modifid Bssl function of first ind, I, of ordr, and Lmma 3 in th intgral (5) abov. Thorm Suppos X is a Mawll random variabl with pdf f X () as givn in () and cdf P( X ) givn by (3) in trms of th F X rror function. Also, suppos Y is a Ric random variabl with pdf f Y (y) givn by (4) in trms of th modifid Bssl function of th first I y. Thn th cdf of Z X Y can b ind prssd as whr F (.) dnots hyprgomtric function of ordr (, ), (s dfinition abov). Using th prssions (3) for cdf of Mawll random variabl X and th prssion

5 SHAKIL & KIBRIA F z /σ a 4 σ σ! + Γ + σ z F 3 a z ;, ; 4 σ + (+ ) Γ a z F + ( + ) 3 a z ;, ; σ az ( a σ) z K + σ whr F (.) dnots hyprgomtric function of ordr (, ), and K (.) dnots th modifid Bssl functions of th scond ind of ordr, (s dfinition abov). Using th prssions (3) for cdf of Mawll random variabl X and th prssion (4) for pdf of Ric random variabl Y, th cdf F z Pr X Y z can b prssd as Fz z Pr X Y z FX f Y(y)dy y az rf /σ y y / σ vy y I dy... az σ a z σ y y (6) whr y >, z >, a >, σ >, v. Th proof of Thorm asily follows by using th dfinition () of modifid Bssl function of th first ind, I, of ordr, substituting y in th first trm and y u in th scond t trm of th intgral (6) abov, and thn using Lmmas and 4 rspctivly. PDF of th Product Z X Y, and th Momnt of RV Z X Y In what follows, without loss of gnrality, for simplicity of computations, this sction discusss th drivation of th pdf of th product Z X Y, whn X and Y ar Ric and Mawll random variabls distributd according to (4) and (), rspctivly, and indpndntly of ach othr. An plicit prssion for th pdf of th product Z X Y in trms of th modifid Bssl function of th scond ind K () has bn drivd in Thorm 3. Th prssion for th th momnt of RV Z X Y in trms of gamma functions has bn drivd in Thorm 4. Thorm 3 Suppos X and Y ar Ric and Mawll random variabls having pdf givn by (4) and (), rspctivly. Thn th pdf of Z X Y can b prssd as /σ fz ( z) n n+ a az K n+ n + n σ ( n! ) σ (7) whr K (.) dnots th modifid Bssl n + functions of th scond ind of ordr (s dfinition abov). n +,

6 ON THE PRODUCT OF MAXWELL AND RICE RANDOM VARIABLES as f Z Th pdf of ( z) z fx Y y y 3 a /σ z a y σ y Z X Y can b prssd f (y) dy z σ vz I dy,... σ y (8) whr y >, z >, a >, σ >, v. Th proof of Thorm 3 asily follows by using th dfinition () of modifid Bssl function of th first ind, I, of ordr, substituting y, and thn using Lmma 4 in th intgral t (8) abov. Thorm 4 If Z is a random variabl with pdf givn by (7), thn its th momnt can b prssd as n E Z /σ 3 n + n a n+ + n+ 3 Γ n Γ σ 4 4 ( n! ) /σ E( Z ) n n+ a az z K n dz... + n + n ( n! ) σ σ (9) By using th quation (6.6.3 / pag 7) from Gradshtyn and Ryzhi (), in th intgral (9) abov, th rsult of Thorm 4 asily follows. Conclusion This articl has drivd th act distributions of th product of two indpndnt random variabls X and Y, whr X and Y hav Mawll and Ric distributions rspctivly. Th pdf and th momnt of th product of two variabls ar also givn. Th distribution is obtaind as a function of hyprgomtric of ordr (, ), whr as th pdf has bn obtaind as a function of Bssl of th scond ind. W hop th findings of th articl will b usful for th practitionrs which ar indicatd in th introduction of th articl. Rfrncs Abramowitz, M., & Stgun, I. A. (97). Handboo of mathmatical functions, with formulas, graphs, and mathmatical tabls. Dovr, N.Y. Andrws, G. E., Asy, R., & Roy, R. (999). Spcial functions. Cambridg: Cambridg Univrsity Prss. Bhargava, R. P., & Khatri, C. G. (98). Th distribution of product of indpndnt bta random variabls with application to multivariat analysis. Annals of th Institut of Statistical Mathmatics, 33, Bohr, H., & Mollrup, I. (9). Lorbog I matmatis Analys, 3. Kopnhagn. Cigizoglu, H. K., & Bayazit, M. (). A gnralizd sasonal modl for flow duration curv. Hydrological Procsss, 4, Galambos, J., & Simonlli, I. (5). Products of random variabls:applications to problms of physics and to arithmtical functions. Boca Raton / Atlanta: CRC Prss. Gradshtyn, I. S., & Ryzhi, I. M. (). Tabl of intgrals, sris, and products (6th d.). San Digo: Acadmic Prss. Grubl, H. G. (968). Intrnationally divrsifid portfolios: Wlfar gains capital flows. Amrican Economic Rviw, 58,

7 SHAKIL & KIBRIA Ladarl, M., Jnsn, V., & Nilsn, B. (997). Total numbr of cancr cll nucli and mitoss in brast tumors stimatd by th optical disctor. Analytical and Quantitativ Cytology and Histology, 9, Mali, H. J., & Trudl, R. (986). Probability dnsity function of th product and quotint of two corrlatd ponntial random variabls. Canadian Mathmatical Bulltin, 9, Nadarajah, S. (5). On th product and ratio of Laplac and Bssl random variabls. Journal of Applid Mathmatics, 4, Nadarajah, S., & Kotz, S. (5). On th product and ratio of Parson Typ VII and Laplac random variabls. Austrian Journal of Statistics, 34(), 3. Prudniov, A. P., Brychov, Y. A., & Marichv, O. I. (986). Intgrals and sris (Volums,, & 3). Amstrdam: Gordon and Brach Scinc Publishrs. Rathi, P. N., & Rohrr, H. G. (987). Th act distribution of products of indpndnt random variabls. Mtron, 45, Roach, M., & Klijunas, P. (97). Bhavior as a function of attitud-toward-objct and attitud-toward-situation. Journal of Prsonality and Social Psychology,, 94. Springr, M. D. & Thompson, W. E. (97). Th distribution of products of bta, gamma and Gaussian random variabls. SIAM Journal on Applid Mathmatics, 8, Stuart, A. (96). Gamma-distributd products of indpndnt random variabls. Biomtria, 49, Wallgrn, C. M. (98). Th distribution of th product of two corrlatd t variats. Journal of th Amrican Statistical Association, 75, 996.