Distribution of the Ratio of Normal and Rice Random Variables

Transcription

1 Journal of Modern Applied Statistical Methods Volume 1 Issue Article Distribution of the Ratio of Normal and Rice Random Variables Nayereh B. Khoolenjani Uniersity of Isfahan, Isfahan, Iran, [email protected] Kaoos Khorshidian Shiraz Uniersity, Shiraz, Iran Follow this and additional wors at: Recommended Citation Khoolenjani, Nayereh B. and Khorshidian, Kaoos (013) "Distribution of the Ratio of Normal and Rice Random Variables," Journal of Modern Applied Statistical Methods: Vol. 1: Iss., Article 7. Aailable at: This Emerging Scholar is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized administrator of DigitalCommons@WayneState.

2 Journal of Modern Applied Statistical Methods Noember 013, Vol. 1, No., Copyright 013 JMASM, Inc. ISSN Emerging Scholars: Distribution of the Ratio of Normal and Rice Random Variables Nayereh B. Khoolenjani Uniersity of Isfahan Isfahan, Iran Kaoos Khorshidian Shiraz Uniersity Shiraz, Iran The ratio of independent random ariables arises in many applied problems. The distribution of the ratio X is studied when X and are independent Normal and Rice random ariables, respectiely. Ratios of such random ariables hae extensie applications in the analysis of noises in communication systems. The exact forms of probability density function (PDF), cumulatie distribution function (CDF) and the existing moments are deried in terms of seeral special functions. As a special case, the PDF and CDF of the ratio of independent standard Normal and Rayleigh random ariables hae been obtained. Tabulations of associated percentage points and a computer program for generating tabulations are also gien. Keywords: functions. Normal distribution, Rice distribution, ratio random ariable, special Introduction For gien random ariables X and, the distribution of the ratio X arises in a wide range of natural phenomena of interest, such as in engineering, hydrology, medicine, number theory, psychology, etc. More specifically, Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, inentory ratios in economics are exactly of this type. The distribution of the ratio random ariables (RRV) has been extensiely inestigated by many authors especially when X and are independent and belong to the same family. Various methods hae been compared and reiewed by authors including Pearson (1910), Greay (1930), Marsaglia (1965, 006) and Nadarajah (006). Nayereh B. Khoolenjani is a Ph.D. student in the Department of Statistics. at: [email protected]. Kaoos Khorshidian is in the Department of Statistics. 436

3 KHOOLENJANI & KHORSHIDIAN The exact distribution of X is deried when X and are independent random ariables (RVs) haing Normal and Rice distributions with parameters ( µσ, ) and ( λ, ), respectiely. The Normal and Rice distributions are well nown and of common use in engineering, especially in signal processing and communication theory. In engineering, there are many real situations where measurements could be modeled by Normal and Rice distributions. Some typical situations in which the ratio of Normal and Rice random ariables appear are as follows. In the case that X and represent the random noises corresponding to two signals, studying the distribution of the quotient X is of interest. For example in communication theory it may represent the relatie strength of two different signals and in MRI, it may represent the quality of images. Moreoer, because of the important concept of moments of RVs as magnitude of power and energy in physical and engineering sciences, the possible moments of the ratio of Normal and Rice random ariables hae been also obtained. Applications of Normal and Rice distributions and the ratio RVs may be found in Rice (1974), Helstrom (1997), Karagiannidis and Kotsopoulos (001), Salo, et al. (006), Withers and Nadarajah (008) and references therein. The probability density function (PDF) of a two-parameter Normal random ariable X can be written as: 1 1 fx ( x) = exp{ ( x µ ) }, x πσ σ < < (1) where < µ < is the location parameter and σ > 0 is the scale parameter. For µ = 0 and σ = 1, (1) becomes the distribution of standard Normal random ariable. A well nown representation for CDF of X is as 1 x µ FX ( x) = 1 + erf ( ) σ () where erf () denotes the error function that is gien by x u = (3) erf ( x) e du π 0 437

4 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES Also, = µ 1 σ j= 0 ( j)! j! µ j EX ( )! ( ). (4) If has a Rice distribution with parameters ( λ, ), then the PDF of is as follows: y ( y + ) y 0 f ( y) = exp{ } Ι ( ), y > 0, 0, λ > 0 (5) λ λ λ where y is the signal amplitude, Ι (.) 0 is the modified Bessel function of the first ind of order 0, λ is the aerage fading-scatter component and is the lineof-sight (LOS) power component. The Local Mean Power is defined as Ω= λ + which equals EX ( ), and the Rice factor K of the enelope is defined as the ratio of the signal power to the scattered power, i.e., K = λ. When K goes to zero, the channel statistic follows Rayleigh distribution, whereas if K goes to infinity, the channel becomes a non-fading channel. For = 0, the expression (5) reduces to a Rayleigh distribution. Notations and Preliminaries Recall some special mathematical functions, these will be used repeatedly throughout this study. The modified Bessel function of first ind of order, is 1 ( x ) 1 I ( ) ( ) 4 x = x (6) (!) Γ ( + + 1) = 0 The generalized hypergeometric function is denoted by F ( a, a,..., a ; b, b,..., b ; z) p q 1 p 1 q ( a ) ( a )...( a ) z ( b) ( b )...( b )! 1 p = (7) = 0 1 q 438

5 KHOOLENJANI & KHORSHIDIAN The Gauss hypergeometric function and the Kummer confluent hypergeometric function are gien, respectiely, by ()() a b z F( abcz, ; ; ) = (8) () c! 1 = 0 and ( a) z F( abz ; ; ) = (9) ( b)! 1 1 = 0 where ( a ), ( b ) represent Pochhammer s symbol gien by Γ ( α + ) ( a) = aa ( + 1) ( a+ 1) =. Γ ( α) The parabolic cylinder function is z ( ) (, ; ) D z = e Ψ z (10) where Ψ ( acz, ; ) represents the confluent hypergeometric function gien by 1 c c 1 1 c Ψ ( acz, ; ) =Γ 1F1( acz ; ; ) +Γ 1F1(1 + a c; cz ; ) 1+ a c a, in which Γ( ai ) a1,..., am i= 1 Γ n b1,..., b =. n Γ( b ) m j= 1 j The complementary error function is denoted by 439

6 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES u = (11) erfc( x) e du π x The following lemmas are of frequent use. Lemma 1 (Equation ( ), Prudnio, et al., 1986). For Re p > 0, Re( α + ) > 0; arg c < π 0 α 1 px x e I ( cx) dx ( α + ) ( α + ). 1 α + c = cp Γ 1F1( ; + 1; ) 4p + 1 Lemma (Equation (.8.9.), Prudnio, et al., 1986). For Re p > 0 ; arg c π < 4 0 n+ 1 px erf ( cx + b) 0 n! ( 1) x e dx = ± n+ 1 erfc( cx + b 1 p n 1 c pb bc erf ( b) + exp( ) erfc( ). n P p c p p c + p + c + p n Lemma 3 (Equation (3.46.1), Gradshteyn & Ryzhi, 000). For Re β > 0, Re > γ γ x exp( { βx γx} dx = ( β) Γ( )exp( ) D ( ). 8β β 440

7 KHOOLENJANI & KHORSHIDIAN The Ratio of Normal and Rice Random Variables The explicit expressions for the PDF and CDF of X are deried in terms of the Gauss hypergeometric function. The ratio of standard Normal and Rayleigh RVs is also considered as a special case. Theorem 1: Suppose that X and are independent Normal and Rice random ariables with parameters ( µσ, ) and ( λν, ), respectiely. The PDF of the ratio random ariable T = X can be expressed as f() t = gt () + g( t), where e gt () = µ µ t λ { + } λ σ 4 σ ( λ t + σ ) σ λ = 0 3 π( λ t + σ ) ( ) Γ + (!) 4 λ µλ t ( 3) D ( + 3) ( ). σ λ t + σ (1) Theorem 1 Proof: f(t) = yf (ty)f (y)dy+ yf ( ty)f (y)dy X X y ( y + ) y = y exp ( ty µ ) exp I 0( )dy 0 πσ σ λ λ λ 1 1 y ( y + ) y + y exp ( ty µ ) exp I 0( )dy 0 πσ σ λ λ λ (13) The two integrals in (13) can be calculated by direct application of Lemma 3. Thus the result follows. Remar : By using expression (10), elementary forms for () gt in Theorem 1 can be deried as follows: 441

8 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES 1 ( σ + µ λ ) σ λ ( ) Γ ( + 3) e λσ 4 31 t gt () λ + µ λ = (, ; ) 3 Ψ (14) + 3 = 0 σ ( t λ + σ ) π( t λ + σ ) (!) Corollary 3 Assume that X and are independent standard Normal and Rayleigh random ariables, respectiely. The PDF of the ratio random ariable T = X can be expressed as λ ft () t =, t > 0 3 ( t λ + 1) (15) Theorem 4: Suppose that X and are independent Normal and Rice random ariables with parameters ( µσ, ) and ( λν, ), respectiely. The CDF of the ratio random ariable T = X can be expressed as Ft () = Gt () G( t) where λ ( ) 4 e n! ( 1) G( t ) 4λ µ = { [ λ erf ( ) λ = (!) + ( ) ( ) σ λ λ 3 tλ µ µλ t exp( )erfc( )]}. ( t ) t λ + σ λ + σ σ ( t λ + σ ) (16) Theorem 4 Proof: The CDF Ft ( ) = Pr( X t) can be written as ty µ ty µ F() t = Φ( ) Φ( ) f ( y) dy, σ σ (17) 0 where Φ (.) is the cdf of the standard Normal distribution. Using the relationship Eq. (17) can be rewritten as 1 x Φ ( x) = erfc( ), (18) 44

9 KHOOLENJANI & KHORSHIDIAN 1 µ ty µ + ty F() t = erfc( ) erfc( ) f ( y) dy σ σ 0 1 µ ty y ( y + ) y = ( ) exp 0( ) erfc I dy 0 σ λ λ λ 1 µ + ty y ( y + ) y erfc( ) exp I 0( ) dy. 0 σ λ λ λ (19) The result follows by using Lemma. Corollary 5: Assume that X and are independent Normal and Rice random ariables with parameters (0, σ ) and ( λ,0), respectiely. The CDF of the ratio random ariable T = X is tλ Ft ( ) =, t> 0. t λ + σ (0) Figures (1) and () illustrate possible shapes of the pdf corresponding to (0) for different alues of σ and λ. Note that the shape of the distribution is mainly controlled by the alues of σ and λ. Figure 1 Plots of the pdf corresponding to (0) for λ = 0.5,1,3,5 and σ =

10 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES Figure Plots of the pdf corresponding to (0) for σ = 0.,0.5,1, and λ = 1. K th Moments of the Ratio Random Variable In the sequel, the independence of X and are used seeral times for computing the moments of the ratio random ariable. The results obtained are expressed in terms of confluent hypergeometric functions. Theorem 6: Suppose that X and are independent Normal and Rice random ariables with parameters ( µσ, ) and ( λν, ), respectiely. A representation for the th moment of the ratio random ariable T= X, for <, is: λ [ ] µ σ j ET [ ] =! e Γ( ) 1F1( ;1; ) ( ) (1) λ λ j= 0 ( j)! j! µ Theorem 6 Proof: Using the independency of X and the expected ratio can be written as X 1 ET ( ) = E E( X ) E( ), = () in which 444

11 KHOOLENJANI & KHORSHIDIAN 1 1 y ( y + ) y E( ) = exp I 0( ) dy y λ λ λ (3) 0 By using lemma.1, the integral (3) reduces to λ 1 e + + E( ) = Γ ( ) 1F1( ;1; ) (4) λ ( λ ) The desired result now follows by multiplying (4) and (4). Remar 7: Formula (1), displays the exact forms for calculating ET ( ), which hae been expressed in terms of confluent hypergeometric functions. The deltamethod can be used to approximate the first and second moments of the ratio T= X. In detail, by taing µ X = EX ( ), µ = E ( ) and using the Deltamethod (Casella & Berger, 00) results in: ν λ µ X µ e µ π 3 ν λ 1F1 λ ET ( ) =. (,1; ) For approximating Var( T ) E ( ) = λ + ν. Thus,, first recall that EX ( ) = µ + σ and X µ X Var( X ) Var( ) Var( ), + µ µ X µ which inoles confluent hypergeometric functions, but in simpler forms. Remar 8: The numerical computation of the obtained results in this article entails calculation of special functions, their sums and integrals, which hae been tabulated and aailable in determinds boos and computer algebra pacages (see Greay, 1930; Helstrom, 1997; and Salo, et al. 006 for more details. 445

12 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES Percentiles Table 1. Percentage points of T = X for λ = λ p = 0.01 p = 0.05 p = 0.1 p = 0.9 p = 0.95 p =

13 KHOOLENJANI & KHORSHIDIAN Table. Percentage points of T = X for λ =.6 5. λ p = 0.01 p = 0.05 p = 0.1 p = 0.9 p = 0.95 p = Tabulations of percentage points t p associated with the cdf (0) of proided. These alues are obtained by numerically soling: T = X are t λ p p λ + σ = t p (5) 447

14 DISTRIBUTION RATIO OF NORMAL AND RICE RANDOM VARIABLES Tables 1 and proide the numerical alues of t p for λ = 0.1,0.,...,5 and σ = 1. It is hoped that these numbers will be of use to practitioners as mentioned in the introduction. Similar tabulations could be easily deried for other alues of λσ, and p by using the sample program proided in Appendix A. References Casella, G., & Berger, L. B. (00), Statistical Inference. Duxbury Press. Gradshteyn, I. S., & Ryzhi, I. M. (000).Table of Integrals, Series, and Products. San Diego, CA: Academic Press. Greay R. C. (1930). The frequency distribution of the quotient of two normal ariates. Journal of the Royal Statistical Society. 93, Helstrom, C. (1997). Computing the distribution of sums of random sine waes and the Rayleigh-distributed random ariables by saddle-point integration. IEEE Trans. Commun. 45(11), Karagiannidis, G. K., & Kotsopoulos, S. A. (001). On the distribution of the weighted sum of L independent Rician and Naagami Enelopes in the presence of AWGN. Journal of Communication and Networs. 3(), Marsaglia, G. (1965). Ratios of Normal Variables and Ratios of Sums of Uniform. JASA. 60, 193, 04. Marsaglia, G. (006). Ratios of Normal Variables. Journal of Statistical Software. 16(4), Nadarajah, S. (006). Quotient of Laplace and Gumbel random ariables. Mathematical Problems in Engineering. ol Article ID 90598, 7 pages. Pearson, K. (1910). On the constants of Index-Distributions as Deduced from the Lie Constants for the Components of the Ratio with Special Reference to the Opsonic Index. Biometria. 7(4), Prudnio, A. P., Brycho.A., Mariche O.I. (1986). Integrals and Series,. New or: Gordon and Breach. Rice, S. (1974). Probability distributions for noise plus seeral sin waes the problem of computation. IEEE Trans. Commun Salo, J., EI-Sallabi, H. M., & Vainiainen, P. (006). The distribution of the product of independent Rayleigh random ariables. IEEE Transactions on Antennas and Propagation. 54(),

15 KHOOLENJANI & KHORSHIDIAN Withers, C. S. & Nadarajah, S. (008). MGFs for Rayleigh Random Variables. Wireless Personal Communications. 46(4), Appendix A The following program in R can be used to generate tables similar to that presented in the section headed 'Percentiles.' p=c(0.01,.05,0.1,0.9,0.95,0.99) sig=1 lambda=seq(0.1,5,0.1) ll=length(lambda) mt=matrix(0,nc=length(p),nr=ll) for(i in 1:ll) { lambda=lambda[i] t=p*sig*sqrt(1/(1-p^))/lambda mt[i,]=t } print(mt) 449