Distribution of the ratio of Maxwell and Rice random variables

Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. 13, 623 - 637

DISTRIBUTION OF THE RATIO OF MAXWELL AND RICE RANDOM VARIABLES M. Shakil Department of Mathematics Miami Dade College Hialeah Campus Hialeah, FL 33012, USA E-mail: [email protected] B. M. Golam Kibria Department of Statistics Florida International University University Park Miami, FL 33199, USA E-mail: [email protected] J. N. Singh Department of Mathematics and Computer Science Barry University Miami Shores, FL 33161, USA E-mail: [email protected] Abstract. The distributions of the ratio of independent random variables arise in many applied problems. These have been extensively studied by many researchers. In this paper, X has been derived when X and Y are Maxwell and Rice the distribution of the ratio Y random variables and are distributed independently of each other. The associated pdfs, cdfs, and kth moments have been given.

Mathematics Subject Classification: 33B99, 33C90, 33D90, 33E99, 62E15 Keywords: Gamma function, Hypergeometric function, Maxwelll distribution, Ratios, Rice distribution.

M. Shakil, B. M. Golam Kibria, J. N. Singh

624

1 Introduction X , when X and Y are independent random variables, arise in Y many applied problems of biology, economics, engineering, genetics, hydrology, medicine, number theory, order statistics, physics, psychology, etc, (see, for example, [4], [6], and [8], X among others, and references therein). The distributions of the ratio , when X and Y are Y independent random variables and come from the same family, have been extensively studied by many researchers, (see, for example, [9], [10], [11], [12], [14], [16], and [17], among others,). In recent years, there has been a great interest in the study of the above kind when X and Y belong to different families, (see, for example, [13], and [15], among others). In X this paper, the distributions of the ratio , when X and Y are independent random Y variables having Maxwell and Rice distributions respectively, have been investigated. The organization of this paper is as follows. Section 2 contains the derivation of the cdf of the X X ratio Z = . The pdf and kth moment of the RV Z = have been derived in Sections 3 Y Y and 4 respectively. Some concluding remarks are given in Section 5.

The distributions of the ratio

X involve some special functions, Y which are defined as follows (see, for example, [1], [5], and [18], among others, for details). The series

The derivations of the cdf, pdf, and kth moment of Z =

k ∞ ⎧ (α ) (α ) L (α ) ⎪ 1 k 2 k ⎪ p k z ⎫ ( ) F α , α , L , α ; β , β , L , β ; z = ⎨ ⎬, ∑ p q 1 2 p 1 2 q ( ) ( ) ( ) β β L β k ! ⎪⎭ k =0 ⎪ 1 2 q k k k ⎩

is called a generalized hypergeometric series of order ( p, q ) , where (α ) k and ( β ) k represent Pochhammer symbols. For p = 1 and q = 2 , we have generalized hypergeometric function 1 F2 of order (1, 2) , given by ∞ ⎧ (α 1 )k z k ⎫ ( ) = F α ; β , β ; z ⎬ . For p = 2 and q = 1 , we have generalized ∑⎨ 1 2 1 1 2 k = 0 ⎩ (β 1 )k (β 2 )k k!⎭

hypergeometric function 2 F1 of order (2, 1) , given by

∞ ⎧ (α ) k ( β ) k z k ⎫ ( ) ( ) ( ) F α , β ; γ ; z F α , β ; γ ; z F β , α ; γ ; z ≡ ≡ = ∑ ⎨ (γ ) k! ⎬ . The following series 2 1 k =0 ⎩ k ⎭ is known as degenerate hypergeometric function or confluent hypergeometric function:

MAXWELL AND RICE RANDOM VARIABLES

625

⎧ (α ) k z k ⎫ ⎬ . The confluent hypergeometric function 1 F1 (α ; β ; z ) is a 1 F1 (α ; β ; z ) = ∑ ⎨ k = 0 ⎩ ( β ) k k! ⎭ degenerate form of the generalized hypergeometric function 2 F1 (α , β ; γ ; z ) of order (2, 1) which arises as a solution the confluent hypergeometric differential equation. Note that z 1 F1 (α , β ; z ) = e 1 F1 (β − α , β ; − z ) which is known as Kummer Transformation. Also, we ∞

have F (α , β ; γ ; z ) = (1 − z )

−β

∞

z ⎞ ⎛ α − 1 −t F ⎜β , γ −α ; γ; ⎟ . The integrals Γ(α ) = ∫ t e dt , and z −1⎠ ⎝ 0

x

γ (α , x ) = ∫ t α − 1e −t dt , α > 0 are called (complete) gamma and incomplete gamma functions 0

∞

respectively, whereas the integral Γ(α , x) = ∫ t α −1e −t dt , α > 0 is called the complementary x

incomplete gamma function. For negative values, gamma function can be defined as (−1) n 2 n π 1⎞ ⎛ , where n ≥ 0 is an integer, (see, for example, [2], and [3], Γ⎜ − n + ⎟ = 2 ⎠ 1.3.5. K (2n − 1) ⎝ among others). The function defined

Γ( p ) Γ( q ) , p > 0, q > 0, is known as beta function (or Euler’s Γ( p + q )

1

by B ( p, q ) = ∫ t p (1 − t ) q − 1 dt = 0

function of the first kind). The error function is defined by erf ( x) = the complementary error, erfc( x ) , is defined as erfc( x) =

2

π

∞

∫e

−u 2

2

π

x

−u ∫ e du , whereas 2

0

du = 1 − erf ( x) . The

x

modified Bessel function of first kind, Iν (x) , for a real number ν , is defined by k

⎛1 2⎞ ⎜ x ⎟ ν ∞ ⎛1 ⎞ ⎝4 ⎠ , where Γ(.) denotes gamma function. Also, in terms of Iν ( x) = ⎜ x ⎟ ∑ ⎝ 2 ⎠ k = 0 (k!) Γ(ν + k + 1)

the confluent hypergeometric function 1 F1 , it can be expressed as ν

1 ⎛ x ⎞ −x ⎛1 ⎞ Iν ( x) = ⎜ ⎟ e 1 F1 ⎜ + ν , 1 + 2ν ; 2 x ⎟ . Γ(ν + 1) ⎝ 2 ⎠ ⎠ ⎝2

When ν = 0 , modified Bessel function of first kind, I 0 ( x ) , of order 0 is obtained as follows:


626 ⎛1 2⎞ x ⎟ ∞ ⎜ 4 ⎝ ⎠ I 0 ( x) = ∑ 2 ( k ! ) k =0

k

(1)

1⎞ π ⎛ For Re ⎜ν + ⎟ > 0, arg ( z ) < ; or Re ( z ) = 0 and ν = 0 , we have the modified Bessel 2⎠ 2 ⎝ function of second kind, Kν (x) , of order ν , given by ν

⎛ z⎞ ⎛1⎞ ⎜ ⎟ Γ⎜ ⎟ ∞ 1 ν− 2 2 Kν ( x) = ⎝ ⎠ ⎝ ⎠ ∫ e − z t (t 2 − 1) 2 dt , 1⎞ ⎛ Γ⎜ν + ⎟ 1 2⎠ ⎝

or, for arg ( z ) < we have Kν ( x) =

π 2

, Re ( z 2 ) > 0 , we have Kν ( x) =

1 2

ν ∞

⎛z⎞ ⎜ ⎟ ⎝2⎠

−t −

e

∫ (t )ν

z2 4t

+1

dt . For non-integer ν ,

0

π {I −ν ( x) − Iν ( x)} . The function, denoted by M k , m ( x ) and defined by 2 sin(ν π )

⎛1 ⎞ M k , m ( x ) = e − x / 2 x m + 1 / 2 1 F1 ⎜ + m − k ; 1 + 2m; x ⎟ ⎝2 ⎠ ⎛z⎞ is called Whittaker function. Also, note that M 0 , μ ( z ) = 2 2 μ Γ( μ + 1) z I μ ⎜ ⎟ . ⎝2⎠

The following Lemmas will also be needed in our derivations.

Lemma 1 (Gradshteyn and Ryzhik (2000), [5], Equation (3.381.4), Page 317). For Re (μ ) > 0 , and Re (ν ) > 0 , ∞

∫t 0

ν −1

e − μ t dt =

1

μν

Γ(ν ) .

Lemma 2 (Gradshteyn and Ryzhik (2000), [5], Equation (6.455.2), Page 663). For Re (α + β ) > 0, Re (β ) > 0, and Re (μ + ν ) > 0 ,

∞

μ −1 − β t ∫ t e γ (ν , α t )dt = 0

α ν Γ( μ + ν ) ν (α + β ) μ +ν

2

⎛ α ⎞ ⎟ F1 ⎜⎜1, μ + ν ; ν + 1; α + β ⎟⎠ ⎝


627

Lemma 3 (Prudnikov et al. (1986), Volume 2, [18], Equation (2.8.5.6), Page 104). For Re ( p ) > 0, Re (α ) > −1, arg (c) < ∞

∫t

α −1

e

− pt2

erf (c t ) dt =

0

π

4

,

c

π p (α + 1) / 2

⎛ 1 α +1 3 − c2 ⎞ ⎛ α +1⎞ ⎟⎟ ; ; Γ⎜ ⎟ 2 F1 ⎜⎜ , 2 2 2 p ⎝ 2 ⎠ ⎝ ⎠

Lemma 4 (Prudnikov et al. (1986), Volume 2, [18], Equations (2.10.3.2), Page 150). For Re (α + ν ) > 0, Re ( p ) > 0, Re (ν ) > 0 and Re (c ) > 0 , ∞

α −1 − p x ∫ x e γ (ν , c x ) dx = 0

c ν Γ(α + ν ) ν ( p) α + ν

2

⎛ c⎞ F1 ⎜⎜ν , α + ν ; ν + 1; − ⎟⎟ p⎠ ⎝

Lemma 5 (Gradshteyn and Ryzhik (2000), [5], Equation (6.643.2), Page 720). 1⎞ ⎛ For Re ⎜ μ + ν + ⎟ > 0 , 2⎠ ⎝

∞

∫x 0

μ −

1 2

e

−α x

I 2ν

1⎞ ⎛ Γ⎜ μ + ν + ⎟ 2 ⎠ − 1 β 2 / 2α − μ 2 β x dx = ⎝ β e α M − μ, ν Γ(2ν + 1)

(

)

⎛β2 ⎜⎜ ⎝α

⎞ ⎟⎟ ⎠

where Iν (.) denotes modified Bessel function of the first kind, and M k , m (.) denotes Whittaker function, (see definition above).

2 Distribution of the Ratio

X Y

Let X and Y be Maxwell and Rice random variables respectively, distributed independently of each other and defined as follows.

2.1 Maxwell Distribution A continuous random variable X is said to have a Maxwell distribution if its pdf f X (x) and cdf FX ( x) = P( X ≤ x) are, respectively, given by


628

2

f X ( y) =

π

a

3

2

2

−a x 2 2

x e

, x > 0, a > 0

(2)

and

⎛3 1 ⎞ 2γ ⎜ , a x2 ⎟ ⎝2 2 ⎠ FX ( x ) =

π

,

(3)

2 ⎛ a ⎞ 2a − a x 2 = erf ⎜⎜ x ⎟⎟ − xe π ⎝ 2 ⎠

where γ (a, x ) and erf ( x ) denote incomplete gamma and error functions respectively, (see definition above).

2.2 Rice Distribution A continuous random variable Y is said to have a Rice distribution if its pdf f Y ( y ) is given by f Y ( y) =

y

σ

2

e−(y

2

+ v 2 ) / 2σ 2

⎛ yv ⎞ I 0 ⎜ 2 ⎟, ⎝σ ⎠

y > 0, σ > 0, v ≥ 0

(4)

where I 0 ( y ) denotes the modified Bessel function of the first kind, (see definition above). For v = 0 , the expression (4) reduces to a Rayleigh distribution. In what follows, we X , when X and Y are Maxwell Y and Rice random variables respectively, distributed independently of each other and defined X as above. An explicit expression for the cdf of in terms of the generalized Y hypergeometric function 2 F1 has been derived in Theorem 2.1. In Theorem 2.2, another explicit expression for the cdf

consider the derivation of the distribution of the product

X in terms of the generalized hypergeometric function Y M k , m , has been derived.

of

2

F1 , and Whittaker function


629

Theorem 2.1 Suppose X is a Maxwell random variable with pdf f X (x) as given in (2) and cdf FX ( x) = P( X ≤ x) given by (3) in terms of the incomplete gamma function. Also, suppose

Y is a Rice random variable with pdf f Y ( y) given by (4) in terms of the modified Bessel function of the first kind I 0 ( y ) . Then the cdf of Z =

⎡ 4 a 3 / 2 σ 3 z 3 e −ν F (z ) = ⎢ 3 π ⎢⎣

2

/ 2σ

2

⎤ ⎥ ⎥⎦

⎧⎛ ν 2 ⎪ ⎜⎜ 2 ∞ ⎪⎝ 2σ ∑⎨ k =0 ⎪ ⎪ ⎩

X can be expressed as Y

⎫ ⎪ ⎛3 5 5 − a z 2 ⎞⎪ ⎜ ,k+ ; ; ⎟⎬ 2 F1 ⎜ 2 2 σ 2 ⎟⎠⎪ ⎝2 ⎪ ⎭

k

⎞ ⎛ 5⎞ ⎟⎟ Γ⎜ k + ⎟ 2⎠ ⎠ ⎝ 2 (k !)

where 2 F1 (.) denotes the generalized hypergeometric function of order (2, 1) , (see definition above).

Proof Using the expressions (3) for cdf of Maxwell random variable X and the expression (4) for ⎛ X ⎞ pdf of Rice random variable Y , the cdf F ( z ) = Pr ⎜⎜ ≤ z ⎟⎟ can be expressed as ⎝Y ⎠ ∞

F ( z ) = Pr ( X ≤ z Y ) = ∫ FX ( z y ) f Y ( y ) dy 0

⎡ 2 e −ν / 2 σ =⎢ 2 ⎢⎣ π σ 2

2

⎤ ⎥ ⎥⎦

∞

−y ∫ ye 0

2

/ 2σ 2

⎛3 1 ⎝2 2

⎞ ⎠

⎛vy 2 ⎝σ

γ ⎜ , a z 2 y2 ⎟ I0 ⎜

⎞ ⎟ dy ⎠

(5)

where y > 0, z > 0, a > 0, σ > 0, v ≥ 0 . The proof of Theorem 2.1 (i) easily follows by substituting y 2 = t , using the Definition (1) of modified Bessel function of first kind, I 0 ( x) , of order 0 , and Lemma 4 in the integral (5) above.

Theorem 2.2 Suppose X is a Maxwell random variable with pdf f X (x) as given in (2) and cdf

FX ( x) = P( X ≤ x) given by (3) in terms of the error function. Also, suppose Y is a Rice random variable with pdf f Y ( y ) given by (4) in terms of the modified Bessel function of the X first kind I 0 ( y ) . Then the cdf of Z = can be expressed as Y


630 ⎧ ⎪⎪ 2 a σ z e −ν 2 / 2 σ 2 F (z ) = ⎨ π ⎪ ⎪⎩

−

3⎞ ⎛ Γ⎜ k + ⎟ 2⎠ ⎝ ∑ 2 (k !) k =0 ∞

2a σ 2 ze

−

⎛ν ⎜⎜ 2 ⎝σ

⎞ ⎟⎟ ⎠

ν 2 ( 2 a σ 2 z 2 +1) 4 σ 2 ( a σ 2 z 2 +1)

ν (a σ z + 1) 2

2

2

k

⎫ ⎪ 3 3 ⎛1 ⎞⎪ , k + ; ; − a σ 2 z 2 ⎟⎬ 2 F1 ⎜ 2 2 ⎝2 ⎠⎪ ⎪⎭

⎛ ν2 M −1, 0 ⎜⎜ 2 2 2 ⎝ 2 σ ( a σ z + 1)

where 2 F1 (.) denotes the generalized hypergeometric function of order M k , m (.) denotes Whittaker function, (see definition above).

⎞ ⎟⎟ ⎠

(2, 1) , and

Proof Using the expressions (3) for cdf of Maxwell random variable X and the expression (4) for ⎛ X ⎞ ≤ z ⎟⎟ can be expressed as pdf of Rice random variable Y , the cdf F (z ) = Pr ⎜⎜ ⎝Y ⎠ ∞

F ( z ) = Pr ( X ≤ z Y ) = ∫ FX ( z y ) f Y ( y ) dy 0

⎡ e −ν / 2 σ =⎢ 2 ⎢⎣ σ 2

2

⎤ ∞ − y 2 / 2σ 2 ⎥∫ ye ⎥⎦ 0

⎧⎪ ⎨erf ⎪⎩

− ⎞ ⎛ a 2a ⎟− ⎜ z y z y e ⎟ ⎜ 2 π ⎠ ⎝

a z2 y2 2

⎫⎪ ⎛ v y ⎬I 0 ⎜ 2 ⎪⎭ ⎝ σ

⎞ ⎟ dy ⎠

(6)

where y > 0, z > 0, a > 0, σ > 0, v ≥ 0 . The proof of Theorem 2.2 easily follows by substituting y 2 = u , using the Definition (1) of modified Bessel function of the first kind, I 0 ( x) , of order 0 , and then using Lemmas 3 and 5 respectively in the integral (6) above.

Corollary 2.1 Using Lemma 2 in the integral (5) above, the cdf of Z = expressed in the equivalent form as

X in Theorem 2.1 can be easily Y

MAXWELL AND RICE RANDOM VARIABLES ⎡ 4 a 3 / 2 σ 3 z 3 e −ν F (z ) = ⎢ 3 π ⎢⎣

2

631

⎤ ⎥ ⎥⎦

/ 2σ 2

⎧⎛ ν 2 ⎪ ⎜⎜ 2 ∞ ⎪ 2σ × ∑ ⎨⎝ k =0 ⎪ ⎪ ⎩

⎞ ⎛ ⎛ 5⎞ 5 5 a σ 2 z2 ⎞⎫ ⎟⎟ Γ⎜ k + ⎟ 2 F1 ⎜⎜1, k + ; ; ⎟⎪ 2⎠ 2 2 a σ 2 z 2 + 1 ⎟⎠ ⎪ ⎠ ⎝ ⎝ ⎬ 5 ⎪ (k !)2 (a σ 2 z 2 + 1)k + 2 ⎪ ⎭ k

Corollary 2.2 Using the definition of Whittaker function, ⎛1 ⎞ M k , m ( x) = e − x / 2 x m + 1 / 2 1 F1 ⎜ + m − k ; 1 + 2m; x ⎟ , ⎝2 ⎠

the cdf of Z =

X in Theorem 2.2 can be easily expressed in the equivalent form as Y

⎛ 2 a σ z e −ν F (z ) = ⎜ ⎜ π ⎝

2

/ 2σ 2

⎧⎛ ν 2 ⎪⎜ ⎞ ∞ ⎪ ⎜⎝ σ 2 ⎟ ∑⎨ ⎟ k =0 ⎠ ⎪ ⎪ ⎩

k ⎫ ⎞ ⎛ 3 1 3 3 ⎟⎟ Γ⎜ k + ⎞⎟ 2 F1 ⎛⎜ , k + ; ; − a σ 2 z 2 ⎞⎟ ⎪ 2⎠ 2 2 ⎝2 ⎠⎪ ⎠ ⎝ ⎬ 2 (k !) ⎪ ⎪ ⎭

ν ⎧ − 2σ 2 ⎪⎪ a σ z e −⎨ 3 ⎪ (a σ 2 z 2 + 1) 2 ⎪⎩ 2

⎫ ⎛3 ⎞⎪⎪ ν ⎜ ; 1; ⎟⎟⎬ 1 F1 ⎜ 2 2 2 2 2 σ ( a σ z 1 ) + ⎝ ⎠⎪ ⎪⎭ 2

where 2 F1 denotes the generalized hypergeometric function of order (2, 1) , and 1 F1 denotes the generalized hypergeometric function of order (1, 1) , (see definition above).


632

3 PDF of the Ratio

X Y

Z=

In what follows, without loss of generality, for simplicity of computations, this section X , when X and Y are Rice and discusses the derivation of the pdf of the ratio Z = Y Maxwell random variables distributed according to (4) and (2), respectively, and X in terms independently of each other. An explicit expression for the pdf of the ratio Z = Y of the gamma function has been derived in Theorem 3.1. The expression for the kth moment X in terms of beta function has been derived in Theorem 3.2. of RV Z = Y

Theorem 3.1 Suppose X and Y are Rice and Maxwell random variables having pdf given by (4) and (2), X can be expressed as respectively. Then the pdf of Z = Y ⎛ 32 −ν 2 / 2 σ 2 ⎜a e f Z (z ) = ⎜ π ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

5⎞ ⎛ Γ⎜ n + ⎟ν 2 n z 2 n + 1 2⎠ ⎝

∞

∑

n=0

2

n−2

σ

2n−3

(n !)

2

(z

2

+aσ

2

)

n +

5 2

(7)

Proof The pdf of Z =

X can be expressed as Y ∞

f Z ( z ) = ∫ y f X ( z y ) f Y ( y ) dy 0

3 ⎛ ⎜ 2 a 2 −ν 2 / 2 σ 2 =⎜ e 2 ⎜ π σ ⎝

⎞ ⎟ z⎟ ⎟ ⎠

∞

∫y 0

− 4

e

z2 y2 2σ

2

−

a y2 2

⎛vz y ⎞ I 0 ⎜ 2 ⎟ dy , ⎝ σ ⎠

(8)

where y > 0, z > 0, a > 0, σ > 0, v ≥ 0 . The proof of Theorem 3.1 easily follows by using the Definition (1) of modified Bessel function of the first kind, I 0 ( x ) , of order 0 , substituting y 2 = t , and then using Lemma 1 in the integral (8) above.


633

Theorem 3.2 Suppose X and Y are Rice and Maxwell random variables having pdf given by (4) and (2), X can be expressed as respectively. Then the pdf of Z = Y

(

f Z ( z ) = 3 2 a 3 / 2 σ 2 e −ν

2

/ 2σ 2

) e( z

ν 2 z2

4σ 2 ( z 2 + a σ 2 ) 2

⎛ ν 2 z2 ⎜ M − 2, 0 ⎜ 2 2 2 + a σ 2) ⎝ 2σ ( z + a σ )

⎞ ⎟⎟ ⎠

where M k , m (.) denotes Whittaker function, (see definition above).

Proof The pdf of Z =

X can be expressed as Y ∞

f Z ( z ) = ∫ y f X (z y ) f Y ( y ) dy 0

3 ⎛ ⎜ 2 a 2 −ν 2 / 2 σ 2 =⎜ e 2 ⎜ π σ ⎝

⎞ ⎟ z⎟ ⎟ ⎠

∞

∫y

− 4

e

z2 y2 2σ 2

−

a y2 2

0

⎛vz y ⎞ I 0 ⎜ 2 ⎟ dy , ⎝ σ ⎠

(9)

where y > 0, z > 0, a > 0, σ > 0, v ≥ 0 . The proof of Theorem 3.2 easily follows by substituting y 2 = t , and then using Lemma 5 in the integral (9) above.

Corollary 3.1 ⎛1 ⎞ Using the definition M k , m ( x) = e − x / 2 x m + 1 / 2 1 F1 ⎜ + m − k ; 1 + 2m; x ⎟ of Whittaker function, ⎝2 ⎠ X in Theorem 3.2 can be easily expressed in the equivalent form as the pdf of Z = Y

(

f Z ( z ) = 3 a 3 / 2 σ ν e −ν

2

/ 2σ 2

)

z 2 (z + a σ 2 )3/ 2

⎛5 ν 2 z2 ⎜ F ; 1 ; 1 1 ⎜ 2σ 2 ( z 2 + a σ 2 ) ⎝2

⎞ ⎟⎟ ⎠

(10)

where 1 F1 denotes the generalized hypergeometric function of order (1, 1) , (see definition above).


634

4 kth Moment of the RV

Z=

X Y

Theorem 4.1 If Z is a random variable with pdf given by (7), then its kth moment can be expressed as

( )

E Zk

⎞ ⎟ ⎟ ⎟ ⎠

⎛ k2 −ν 2 / 2 σ 2 ⎜ a e =⎜ π ⎜ ⎝

5⎞ ⎛ Γ⎜ n + ⎟ ν 2 n 2⎠ ⎛ 2n + k + 2 3 − k ⎞ ⎝ B⎜ , ⎟, −1 ≤ k < 3 , ∑ 2 n −1 2n −k 2 ⎠ (n !) ⎝ 2 σ n=0 2 ∞

where B ( p, q ), p > 0, q > 0 , denotes Beta function (or Euler’s function of the first kind), (see definition above).

Proof We have

( )

E Zk

⎛ 32 −ν 2 / 2 σ 2 ⎜a e =⎜ π ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

5⎞ ⎛ Γ⎜ n + ⎟ ν 2 n 2⎠ ⎝ ∑ 2n − 3 n −2 (n !)2 σ n=0 2 ∞

∞

k ∫z 0

z 2 n +1

(z

2

+aσ

2

)

n+

5 2

dz

(11)

Substituting z 2 = u , and using the equation (3.194.3 / page 285) from Gradshteyn and Ryzhik, [5], in (11), the result of Theorem 4.1 easily follows, provided − 1 ≤ k < 3 . It is evident from Theorem 4.1 that only the moments of order k = 1 and k = 2 exist and are given by ⎛ 12 −ν 2 / 2 σ 2 ⎜ a e α 1 = E (Z ) = ⎜ π ⎜ ⎝

⎛ ae

α 2 = E (Z 2 ) = ⎜⎜ ⎝

−ν 2 / 2 σ 2

π

⎞ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

5⎞ ⎛ Γ⎜ n + ⎟ ν 2 n 2⎠ ⎛ 2n + 3 ⎞ ⎝ B⎜ , 1⎟ , and ∑ 2 n −1 2 n −1 (n !) ⎝ 2 σ ⎠ n=0 2 ∞

5⎞ ⎛ Γ⎜ n + ⎟ ν 2 n 2⎠ ⎛ ⎝ B ⎜ n + 2, ∑ 2 n −1 2n −2 (n !) ⎝ σ n=0 2 ∞

1⎞ ⎟. 2⎠


635

Using the above expressions for α1 and α 2 , one can easily determine the variance given by

β 2 = Var Z = α 2 − α 1 2 . Further, the first negative moment of Z (by taking k = −1 in Theorem 4.1) is given by

( )

E Z −1

−ν 2 / 2 σ 2 ⎛ 1 ⎞ ⎛⎜ e = E⎜ ⎟ = aπ ⎝ Z ⎠ ⎜⎝

⎞ ⎟ ⎟ ⎠

5⎞ ⎛ Γ⎜ n + ⎟ ν 2 n 2⎠ ⎛ 2n + 1 ⎞ ⎝ B⎜ , 2⎟ . ∑ 2 n −1 2n + 1 (n !) ⎝ 2 σ ⎠ n=0 2 ∞

For a discussion on the existence of the first negative moment of a continuous random variable and its applications, the interested readers are referred to [7, Section 6.9.1, P. 242] and references therein.

Theorem 4.2 If Z is a random variable with pdf given by (10), then its kth moment can be expressed as

( )

E Z

k

k +2 ⎛3 2 2 ⎞ ⎛k +2 5 3 ν2 ⎛ k + 2 1− k ⎞ k , , ; , 1; = ⎜⎜ ν σ a 2 e −ν / 2 σ ⎟⎟ B ⎜ ⎟ 2 F2 ⎜⎜ 2 ⎠ 2 2 2σ 2 ⎝ 2 ⎠ ⎝ 2 ⎝2

⎞ ⎟⎟ , ⎠

where − 2 < k < 1, and B ( p, q ), p > 0, q > 0 , denotes Beta function (or Euler’s function of the first kind) and 2 F2 denotes the generalized hypergeometric function of order ( 2, 2) , (see definition above).

Proof We have

( ) (

E Z k = 3 a 3 / 2 σ ν e −ν

2

/ 2σ 2

)∫z ∞

0

k

⎛5 ν 2 z2 ⎜ F ; 1 ; 1 1 ⎜ 2σ 2 ( z 2 + a σ 2 ) ⎝2 (z 2 + a σ 2 )3/ 2

⎞ ⎟⎟ ⎠ z dz

(12)

Substituting z 2 = u , and using the equation (2.22.2.2) / page 335) from Prudnikov et al. (1986), Volume 3, [18], in (12), the result of Theorem 4.2 easily follows, provided − 2 < k < 1 . It is evident that only the first negative moment of Z can be obtained from Theorem 4.2 by taking k = −1. This is given by


636

( )

E Z

−1

1 / 2 −ν ⎛ 1 ⎞ ⎛ 3ν a e = E⎜ ⎟ = ⎜ 2σ ⎝ Z ⎠ ⎜⎝

2

/ 2σ 2

2 ⎞ ⎛1 ⎞ ⎛ ⎟ B ⎜ , 1⎟ 2 F2 ⎜ 1 , 5 ; 3 , 1; ν ⎜2 2 2 ⎟ ⎝2 ⎠ 2σ 2 ⎝ ⎠

⎞ ⎟⎟ . ⎠

5 Concluding Remarks This paper has derived the distribution of the ratio of two independent random variables X and Y , where X has Maxwell and Y has Rice distribution. The pdf and kth moment of the ratio of two variables are also given. The distribution is obtained as a function of hypergeometric and Whittaker functions, where as the pdf has been obtained as a function of gamma, Whittaker, and hypergeometric functions. We hope the findings of the paper will be useful for the practitioners that have been mentioned in Section 1.

Acknowledgment The authors would like to thank the editor and referee for their useful comments and suggestions which considerably improved the quality of the paper.

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