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On the Cumulative Distribution Function of the Difference of Two Dependent Chi Square

Random Variables

Elias G. Saleeby1,∗, Anwar H. Joarder2, Nima Rabiei3
1Dept. of Mathematics, Al Akhawayn University (AUI), Ifrane 53000, Morocco

esaleeby@yahoo.com, e.saleeby@aui.ma
2Dept. of Mathematics, Al Akhawayn University (AUI), Ifrane 53000, Morocco

ajstat@gmail.com
3Dept. of Engineering and Natural Sciences, International University of Sarajevo (IUS), Sarajevo, BIH

nrabiei@ius.edu.ba
∗Correspondence: esaleeby@yahoo.com, e.saleeby@aui.ma

Abstract. In this article, we reexamine the derivation of the cumulative distribution function of thedifference of two dependent chi-square random variables with the same degrees of freedom. We derivethe cdf for this difference for even degrees of freedom and discuss a discrepancy that we have foundwith a reported cdf of this difference for even degrees of freedom in [6]. For odd degrees of freedom,an expression for the cdf seems to be unknown. In this case, we derive a representation of the cdfin terms of the Meijer G-function. These representations allowed us to compute percentiles for evenand odd degrees of freedom.

1. Introduction
In the algebra of random variables, finding the probability density function (pdf) and the cumulativedistribution function (cdf) of the difference and the sum of two random variables (rvs) are standardproblems. It is well known that such combinations of rvs appear within the theory of statisticsand in its applications. It is also clear that when the two rvs are dependent, the analysis of theproblem is more technically complicated. In particular, in this note we focus mainly on deriving thecdf for the difference of two dependent central chi-square random variables with the same numberof degrees of freedom. Results on this problem seem to have been around for a while and arereported in some detail in [6]. Not aware initially of the results in [6], we have carried out the basicanalysis and derived the cdf. The cdf expressions which we have obtained appear in different formsthan those reported in [6]. In an attempt to see how these different representations correspond,we discovered a discrepancy between the two forms of the cdfs. In the analysis below we give

Received: 2 Apr 2025.
Key words and phrases. cumulative distribution; chi square; difference of random variables.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 2a modification to the cdf expression for even degrees of freedom reported in [6]. The cdf for thedifference of dependent central chi square rvs with odd degrees of freedom seems to be unknown,and it is not reprted in [6]. A representation of the cdf in terms of a Meijer G-functions seems to fillin this gap. The article is organized as follows. We first discuss a bivariate chi square distributionof the Kibble-type on which the rest of the analysis is based. Then we present and discuss thepdf of the difference of two dependent chi square rvs, followed by a derivation of the cdf of thisdifference. We end the article by computing a sampling of percentiles from the cdf expressions forboth even and odd degrees of freedom.
2. Density Functions

In this section, we examine a joint pdf for correlated gamma rvs derived by W.F. Kibble in [2], obtainfrom it the joint pdf of two dependent central chi square rvs, forms an initial reference point forfurther analysis. This joint pdf turns out to be identical to the joint pdf reported in [6]. We thenobtain the pdf for the difference of two dependent central chi square random variables and comparewith the piecewise given expressions of the pdf reported in [6].Given a vector (X1, · · · , Xn) of Gaussian rv’s with zero mean, then for n > 1, Y =
∑

limn
i=1X

2
iis a central chi square rv with n degrees of freedom. For simplicity, we assume that the rvs Xi arestandard normal. Take two such vectors Y1 and Y2., their joint probability density function (pdf) isgiven by (e.g., see [6], p. 21)

p
Y1,Y2

(y1, y2) =
(y1y2)

1
2 ( n2−1)

4Γ
(
n
2

)
(1− ρ2) (2 |ρ|)

n
2
−1

exp

(
−

y1 + y2

2
√

1− ρ2

)
I n

2
−1

(
|ρ| √y1y2

1− ρ2

)
, (1)

where y1, y2 ≥ 0, −1 < ρ < 1, ρ 6= 0 (if n > 2), and Iν is the modified Bessel function of the firstkind of order ν. As the
lim
ρ→0

I n
2
−1

(
|ρ|√y1y2

1−ρ2

)
|ρ|

n
2
−1

=
(y1y2)

n−2
4

2
n
2
−1Γ

(
n
2

) ,
the joint pdf reduces to the product of two univariate chi square pdfs.
Remark 1. Recall that the pdf of the unscaled univariate gamma distribution is given by
f (x ;α, β) = xα−1e

− x
β

Γ(α)βα , where α > 0, β > 0, and x ≥ 0. Putting α = n
2 , β = 2, one obtains

the univariate chi square distribution as a special case. It is well known that there are different
variants of the bivariate gamma pdf’s in the literature (e.g., see [3], Ch. 48). Among the earliest
bivariate pdfs for the gamma distribution is that of Kibble [2] derived for the case β = 1 (scaled
gamma’s). Adjusting Kibble’s bivariate gamma for the scale parameter β = 2, with the shape
parameter α = n

2 , one obtains the pdf given in (1) - which is a Kibble type bivariate chi square
pdf. Adjusting the scale parameter, one would obtain the pdf with σ2

1 6= 1 6= σ2
2.

Consider now the difference W of two dependent central chi square rvs, X, Y, each having a n = 2m,
m > 1, degrees of freedom (using the notation in [6]). Then it is reported in [6] that the rvW = X−Y

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 3has the following piecewise defined pdf
pW (w) =

|w |m−1

(m − 1)!22m (1− ρ2)
m
2

exp

(
w

2
√

1− ρ2

)
m−1∑
i=0

(m + i − 1)!

i ! (m − i − 1)!

(√
1− ρ2

|w |

)i
, w < 0

pW (w) =
|w |m−1

(m − 1)!22m (1− ρ2)
m
2

exp

(
−

w

2
√

1− ρ2

)
m−1∑
i=0

(m + i − 1)!

i ! (m − i − 1)!

(√
1− ρ2

|w |

)i
, w ≥ 0

Using the following expansion of the Macdonald function [8]
Km− 1

2
(w) =

( π
2w

) 1
2
e−w

m−1∑
i=0

(m + i − 1)!

i ! (m − i − 1)! (2w)i
,

and K−ν(w) = Kν(w), the pdf given in [6] for W can be written as
pW (w) =

|w |m−
1
2

22m
√
π (1− ρ2)

2m+1
4 (m − 1)!

Km− 1
2

(
|w |

2
√

1− ρ2

)
, w 6= 0,

and, pW (0) := limw→0 pW (w), which evaluates to Γ(m− 1
2 )

4
√
π
√

1−ρ2(m−1)!
. The simplest definition of

the Macdonald function is K
m− 1

2

(w) := π(−1)k−1

2

(
I−m+ 1

2
(w)− Im− 1

2
(w)

)
, m integer, (amongother names of Kν , it is often also called the modified Bessel function of the second kind (like inMathematica)). Replacing the m used in ( [6], p. 29 ) by n

2 , where n now is the number of degreesof freedom of the rvs X and Y, then followed by replacing n by m, we obtain
fW (w) =

|w |
m−1

2

2m
√
π (1− ρ2)

m+1
4 Γ

(
m
2

)Km−1
2

(
|w |

2
√

1− ρ2

)
, w ∈ R\{0}. (2)

and fW (0) :=
Γ(m−1

2 )
4
√
π
√

1−ρ2Γ(m2 )
. Clearly fW (w) = fW (−w) . For m = 1, fW (w) =

1

2π
√

1−ρ2
K0

(
|w |

2
√

1−ρ2

)
. For m = 2, as K 1

2
(z) =

(
π
2z

) 1
2 e−z , then fW (w) = 1

4
√

1−ρ2
e
− |w |

2
√

1−ρ2 .These cases match with equations (4.20) and (4.23) given in ( [6], p. 29). For odd degrees offreedom, it is reported in ( [6], p. 30), using his notation for n = n1 = n2 = 2m + 1, and for
σ1 = σ2 = 1, that the pdf is given by

pW (w) =
|w |m

√
πΓ
(
m + 1

2

)
(1− ρ2)

m+1
2

Km

(
|w |

2
√

1− ρ2

)
. (3)

Writing this in terms of n and setting n = m to adjust back to our notation, we see that (3) matcheswith (2). Figure 1 shows a plot of fW (w) .

3. Cumulative Distribution Functions
In this section, we derive representations of the cdf of W for even and odd degrees of freedom. Inthe case of even degrees of freedom, we compare our result with the cdf expression reported in [6].Let c = 16

(
1− ρ2

)
.

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 4

Figure 1. The pdf fW (w) for m = 10, 15, 20, and ρ = 0.7.

Theorem 1. If the symmetric pdf of W is given by (2), then the cdf of W for w > 0 is given by

FW (w) =

∫ 0

−∞
fW (w) dw +

∫ w

0

fW (s) ds =
1

2
+ J. (4)

For m an even positive integer > 3, J is given by

J =
Γ
(
m−1

2

)
w

4Γ
(
m
2

)√
π
√

1− ρ2
1F2

(
1

2
;

3−m
2

,
3

2
;
w2

c

)
+

Γ
(

1−m
2

)
wm

22mmΓ
(
m
2

)√
π (1− ρ2)

m
2

1F2

(
m

2
;
m + 1

2
,
m + 2

2
;
w2

c

)
,

where −1 < ρ < 1, 1F2 (w) =
∑∞
k=0

(a)k
(b1)k(b2)k

wk

k! , and (γ)k = Γ(γ+k)
Γ(γ) , b1, b2 6= 0,−1,−2, · · · .

Proof. First, observe that by the symmetry of the pdf in (2), I = 1
2 . To evaluate J, we use formula03.04.21.0014.01 in [9], namely,∫

zνKν (z) dz =
πz csc (νπ)

2ν+2

 4ν
√
π 1F̃2

(
1
2 ; 1− ν, 3

2 ; z
2

4

)
−z2νΓ

(
ν + 1

2

)
1F̃2

(
ν + 1

2 ; ν + 1, ν + 3
2 ; z

2

4

)  ,
where 1F̃2 (a; b1, b2; z) := 1F2 (a; b1, b2; z) / (Γ (b1) Γ (b2)) is the regularized generalized hyper-geometric function (rhgf). Then, it is straight forward to show that

J =
π csc

(
m−1

2 π
)
w

23Γ
(
m
2

)√
1− ρ2

1F̃2

(
1

2
;

3−m
2

,
3

2
;
w2

c

)
−
√
π csc

(
m−1

2 π
)
wm

22m+1 (1− ρ2)
m
2

1F̃2

(
m

2
;
m + 1

2
,
m + 2

2
;
w2

c

)
,

where,

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 5

1F̃2

(
1

2
;

3−m
2

,
3

2
; z

)
=

∞∑
k=0

1
√
π
(

1
2 + k

)
Γ
(

3−m
2 + k

) zk
k!
,

1F̃2

(
m

2
;
m + 1

2
,
m + 2

2
; z

)
=

∞∑
k=0

1(
m
2 + k

)
Γ
(
m
2

)
Γ
(
m+1

2 + k
) zk
k!

;

are the rhgfs defined for all z ∈ C. Note that for m even, the limz→0 J = 0 = J (0) .Using Euler’s reflection formula (Γ (1− z) Γ (z) = π
sinπz , z = m−1

2 , which is not an integer when
m is even), we obtain

J =
Γ
(

3−m
2

)
Γ
(
m−1

2

)
23Γ

(
m
2

)√
1− ρ2

w 1F̃2

(
1

2
;

3−m
2

,
3

2
;
w2

c

)
−

Γ
(

3−m
2

)
Γ
(
m−1

2

)
22m+1

√
π (1− ρ2)

m
2

wm 1F̃2

(
m

2
;
m + 1

2
,
m + 2

2
;
w2

c

)
.

Then using Γ (1 + z) = zΓ (z) , the J given in the theorem follows. �

Note that the integral I =
∫ w
−∞ fW (w) dw for Re (w) < 0, and even m > 0, evaluates to

I =
1

2
+

Γ
(
m−1

2

)
w

4Γ
(
m
2

)√
π
√

1− ρ2
1F2

(
1

2
;

3−m
2

,
3

2
;
w2

c

)
−

Γ
(

1−m
2

)
(−w)m

22mmΓ
(
m
2

)√
π (1− ρ2)

m
2

1F2

(
m

2
;
m + 1

2
,
m + 2

2
;
w2

c

)
. (5)

Figure 2 shows plots for the cdf FW (w) by Eq.(4).

Figure 2. The cdf FW (w) for m = 4, 16, and ρ = 0.7, 0.9.

The cdf for the difference W of two dependent central chi square random variables with 2m degreesof freedom is reported in ( [6], p. 30) (for σ2
1 = σ2

2 = 1) as

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 6

PW (w) =


(1−ρ2)

m
2

22m(m−1)! exp

(
w

2
√

1−ρ2

)∑m−1
i=0

∑m−i−1
l=0

(m+i−1)!(1−ρ2)
i+l+1

2l+1

i!(m−i−l−1)!l! (−w)m−i−l−1 , w < 0

1− (1−ρ2)
m
2

22m(m−1)! exp

(
− w

2
√

1−ρ2

)∑m−1
i=0

∑m−i−1
l=0

(m+i−1)!(1−ρ2)
i+l+1

2l+1

i!(m−i−l−1)!l! (w)m−i−l−1 , w ≥ 0


Comparing this cdf with the cdf in (4), we have noticed that there were discrepancies between thetwo cdfs. For example, in Figure 3 the plots of the cdf FW (w) and the cdf PW (w) are displayedfor ρ = 0.7, where m = 2 is used in PW (w) , and m = 4 is used in FW (w) (that is, the degreesof freedom equal 4).

Figure 3. Comparison of the cdfs PW (w) and FW (w) for ρ = 0.7 and degrees offreedom 4.

In order to determine the source of this discrepancy, we examine the derivation of PW (w) for w < 0.This is enough, as for a symmetric density function about the y-axis one has F (x) = 1 − F (−x).This formula was employed in [6] for his piecewise presentation of PW (w).

To start with, consider the following integral∫ w

−∞
(−y)m−i−1 e

y
a dy = am−iΓ(m − i ,−

w

a
), Re (w) < 0,

where a = 2
√

1− ρ2, and Γ[n, x ] is the upper (or complementary) incomplete gamma function. Foran integer n, the expansion Γ(n, x) = (n − 1)!e−x
∑n−1
k=0

xk

k! is given in [7]. Therefore, the correctionof the CDF in the notation given in [6], for w < 0, can be written as
PW (w) =

1

2m (m − 1)!
exp

(
w

2
√

1− ρ2

)
m−1∑
i=0

m−i−1∑
l=0

(m + i − 1)!

2i i !l!

(
−w

2
√

1− ρ2

)l
.

The results from this formula matches those obtained from our cdf (5). Furthermore, it is worthnoting that the pdf and the cdf of a sum of dependent rvs X and Y, say V = X + Y, is sometimes

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 7derived indirectly from W = X − Z by setting Y = −Z (e.g., see [6], Ch. 5).
For m odd, we note that no expression for the cdf is given in [6], and to our knowelege it is unknown.In our representation of the cdf in (4), or in its regularized version, we encouter the evaluation of thegamma function at negative integers (poles) where it is not defined. It turns out that for odd m theintegral J can be expressed in terms of the Meijer G-function. Recall, that the Meijer G-functionis defined as a Mellin-Barnes integral (an inverse Mellin transform) (see [1])

Gm,np,q

(
ap

bq
|z, r

)
:=

r

2πi

∫
L

∏m
j=1 Γ

(
bj − r s

)∏n
j=1 Γ

(
1− aj − r s

)∏q
j=m+1 Γ

(
1− bj + r s

)∏p
j=1 Γ

(
aj − r s

)z sds,
where ap = (a1, · · · , ap) and bq = (b1, · · · , bq) , and L is a contour in the complex s-plane withcertain properties (e.g., see [11] for further conditions for which the definition holds). For brevity,we describe this integral in the following remark in a bit more details in our specific case.
Remark 2. Returning to the contour integral representation of Meijer G, but now as given in [10],

we see in our specific case that G2,1
1,3

(
1

1
2

m
2 0

|z

)
= 1

2πi

∫
L

Γ( 1
2

+s)Γ(m2 +s)Γ(−s)

Γ(1+s) z−sds, where

L is a contour in the complex s-plane, which exists as a1 − bi − 1 /∈ N (see [11]). In [10], s is
replaced with −s, and hence the poles and L undergo a reflection. More specifically, a contour L
is chosen to separate the poles of Γ

(
1
2 + s

)
and Γ

(
m
2 + s

)
from those of Γ (−s) . The contour in

our case starts at −∞ encircling the poles of Γ
(

1
2 + s

)
(at s = −1

2 − n, n = 0, 1, 2, · · · ) and those
of Γ

(
m
2 + s

)
(at s = −m2 − n, n = 0, 1, 2, · · · ) but not those of Γ (−s) (at n = 0, 1, 2, · · · ) and

returning to −∞. Since in our case it holds that 0 ≤ m < q and 0 ≤ p < q; and that 1 + 1
2 , 1 + m

2

are not integers whenever m is odd, the integral converges for all z 6= 0; and the Meijer function
is an analytic function except for z = 0 (e.g., see [11]).

Theorem 2. For odd m > 0, the J in (4) is given in trems of the Meijer G-function, and hence, the
cdf can be expressed as

FW (w) =
1

2
+

1

2
√
πΓ
(
m
2

)G2,1
1,3

(
1

1
2

m
2 0

|
w2

c

)
, w > 0. (6)

Proof. The entry 07.34.03.0605.01 in [10] gives a representation of the Bessel K function as
G2,0

0,2

(
−

b1, b2

|z

)
= 2z

b1+b2
2 Kb1−b2

(
2
√
z
)
.

However, the MeijerReduce command in [5] gave us
G2,0

0,2

(
−

b1, b2

|z,
1

2

)
= 2z

b1+b2
2 Kb1−b2

(
2
√
z
)
,

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 8where the 1
2 equals the r in the contour integral in the definition of the Meijer G-function givenabove. Consequently, the integral in terms of the Meijer G-function becomes∫ w

0

x
m−1

2 Km−1
2

(
x

2
√

1− ρ2

)
dx =

∫ w

0

1

2
x
m−1

2 G2,0
0,2

(
−

m−1
4 , 1−m

4

|
x

4
√

1− ρ2
,

1

2

)
dx.

This integral evaluates to a Meijer G-function with r = 1 when an odd positive numerical value of
m is specified. Observing the pattern, we arrive at the expression in (6). By Remark 2, z 6= 0, andas the limit of G2,1

1,3 (z |·) → 0 as z → 0, FW (0) := 1
2 . Moreover, limw→∞ FW (w) = 1. �

Let b = 2
√

1− ρ2, and hence 4b2 = c.The integral I =
∫ w
−∞ fW (w) dw for Re (w) < 0, and odd

m > 2, evaluates to

I =

b
m+1

2

[
2π(m − 2)!!− 2

m+1
2 G3,1

1,2

(
1

1
2 ,
m
2 , 0

| w2

4b2

)]
2m+2

√
π (1− ρ2)

m+1
4 Γ

(
m
2

) , m = 4k − 1;

I =

b
m+1

2 π (m − 2)!! + w
m+1

2 G3,1
1,2

(
3−m

4
1−m

4 , m−1
4 ,−m+1

4

| w2

4b2

)
2m+2

√
π (1− ρ2)

m+1
4 Γ

(
m
2

) , m = 4k + 1,

where k = 1, 2, · · · ; and n!! is the double factorial, which is the product of all positive odd integersup to n.

Figure 4. The cdf FW (w) for m = 11, 18, 25 and ρ = 0.7.
Figure 4 shows plots of the cdf generated using the Meijer function representation of FW (w).

4. Percentiles
In this section, to illustrate the use of the equation FW (w) = 1

2 +J , we compute the 95th percentile
(α = 0.05), for ρ = 0.8, 0.9, 0.95, and for degrees of freedom m = 3, · · · , 30. For m even, using

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 9either (4), or the regularized generalized hypergeometric function representations of J, one obtainsthe percentiles reported in Table 1.
Remark 3. Given that the integral representig G2,1

1,3 converges, L. Slater’s Theorem (e.g., see [11]
for the general statement, [1,4,12]) which gives us an expression of the Meijer G-function in terms
of two generalized hypergeometric functions for bi − bj /∈ Z, i 6= j . In our specific case, using this
theorem gives us that

G2,1
1,3

(
a1

b1 b2 b3

|z

)

=

∏2
j=1 Γ

(
bj − bh=1

)∗∏1
j=1 Γ

(
1 + b1 − aj

)∏
lim3

j=3 Γ
(

1 + b1 − bj
) 1F2 (1 + b1 − a1; 1 + b1 − b2, 1 + b1 − b3; z)

+

∏2
j=1 Γ

(
bj − bh=2

)∗∏1
j=1 Γ

(
1 + b2 − aj

)∏3
j=3 Γ

(
1 + b2 − bj

) 1F2 (1 + b2 − a1; 1 + b2 − b1, 1 + b2 − b3; z) ,

where the * indicates that the term corresponding to j = h is omitted. This equation reduces to

the expression 1
2
√
πΓ(m2 )

G2,1
1,3

(
1

1
2

m
2 0

| w2

16(1−ρ2)

)
= J, where J is as in Theorem 1.

A similar conclusion can be obtained from formula 07.34.03.0727.01 in [10]

G2,1
1,3

(
a1

b1 b2 b3

|z

)

= π csc ((b2 − b1)π)

[
Γ (1− a1 + b1) zb1

1F̃2 (1− a1 + b1; b1 − b2 + 1, b1 − b3 + 1; z)

−Γ (1− a1 + b2) zb2
1F̃2 (1− a1 + b2; 1− b1 + b2, b2 − b3 + 1; z)

]
,

where b2 − b1 /∈ Z. Solving the linear system

1− a1 + b1 =
1

2
, b1 − b2 + 1 =

3−m
2

, b1 − b3 + 1 =
3

2
, 1− a1 + b2 =

m

2
,

1− b1 + b2 =
m + 1

2
, b2 − b3 + 1 =

m + 2

2
;

we obtain that a1 = 1 + b3, b1 = 1
2 + b3, b2 = m

2 + b3. For simplicity, take b3 = 0, and then

we have G2,1
1,3

(
1

1
2

m
2 0

|w2

c

)
, with m−1

2 /∈ Z, w > 0. Furthermore, evaluating the contour

integral described in Remark 2, using formula 07.34.06.0045.01 in [10] for the residues series,
resulted in combination of four 1F2 generalized hypergeometric functions that also blow up for odd
m. Therefore, it appears that in the hypergeometric representation of this Meijer G-function, the
restriction b2 − b1 /∈ Z for odd m cannot be removed.

Clearly, the representation in (6) holds for even m as the condition m−1
2 /∈ Z holds; and thecomputed percentiles from this expression were identical to those reported in Table 1. It turns outthat in Mathematica [5] for the case where m ≥ 1 is odd, it is possible to solve for w > 0 using

https://doi.org/10.28924/ada/ma.5.19


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 10the FindRoot command, which solves numerically for an initial guess at the root. As it is not clearenough how the regularization has occured for odd m, we implemented Newton’s method to findthe root using formula 07.34.20.0001.01 in [10] for the derivative of the Meijer G-function
F ′W (w) =

2w

2c
√
wΓ
(
m
2

)G2,2
2,4

(
−1, 0

−1
2 ,
m
2 − 1, 0,−1

|
w2

c

)
.

Solving (6) by Newton’s method, using positive initial guesses, gave the same values as those
reported in Table 1. Newton’s method worked as an expansion of G2,1

1,3

(
1

1
2

m
2 0

|z

) can be
written as

π sec
(mπ

2

)(
−
√
πz 1F̃2

(
1

2
;

3−m
2

,
3

2
; z

)
+ z

m
2 Γ
(m

2

)
1F̃2

(
m

2
;
m + 1

2
,
m + 2

2
; z

))
.

Then the ratio of this G-function to its derivative is clearly regular for odd m and w > 0, as thesecant function cancels out.
m\ρ 0.8 0.9 0.95 m\ρ 0.8 0.9 0.95
4 3.92617 2.85230 2.04325 3 3.39639 2.46742 1.76754
6 4.81249 3.49619 2.50450 5 4.39180 3.19057 2.28557
8 5.55949 4.03888 2.89325 7 5.19934 3.77723 2.70582
10 6.21796 4.51724 3.23593 9 5.89785 4.28469 3.06934
12 6.81358 4.94995 3.54590 11 6.52251 4.73850 3.39442
14 7.36152 5.34802 3.83106 13 7.09280 5.15280 3.69121
16 7.87168 5.71864 4.09655 15 7.62084 5.53641 3.96601
18 8.35093 6.06681 4.34596 17 8.11482 5.89528 4.22309
20 8.80428 6.39616 4.58189 19 8.58058 6.23365 4.46548
22 9.23552 6.70945 4.80631 21 9.02246 6.55467 4.69544
24 9.64758 7.00881 5.02076 23 9.44379 6.68607 4.91470
26 10.0428 7.29594 5.22645 25 9.84718 7.15381 5.12463
28 10.4231 7.57224 5.42437 27 10.2347 7.43537 5.32633
30 10.7901 7.83883 5.61535 29 10.6082 7.70668 5.52069Table 1. The 95th Percentile (α = 0.05) for 3 ≤ m ≤ 30, and ρ = 0.8, 0.9, 0.95.

References
[1] H. Bateman, A. Erdelyi, Higher Transcendental Functions, Vol I. McGraw Hill, New York (1953).[2] W. F. Kibble, A Two Variate Gamma Type Distribution, Sankhya 5(1941), 137-150. http://www.jstor.org/

stable/25047664.[3] S. Kotz, N. Balakrishna, N. L. Johnson, Continuous Multivariate Distributions, John Wiley, USA (2000).

https://doi.org/10.28924/ada/ma.5.19
http://www.jstor.org/stable/25047664
http://www.jstor.org/stable/25047664


Eur. J. Math. Anal. 10.28924/ada/ma.5.19 11

[4] Y. L. Luke, The Special Functions and Their Approximations, Vol 1. Academic Press, New York (1969). https:
//lib.ugent.be/catalog/ebk01:1000000000789815.[5] A. P. Prudnikov, Y. A. Brychkov, O. I. Marchev, Integrals and Series Vol 3, More Special Functions, Gordan andBreach, New York (1990).[6] M. k. Simon, Probability Distributions Involving Gaussian Random Variables, Springer, US (2002).[7] E. W. Weisstein, Incomplete Gamma Function. https://mathworld.wolfram.com/IncompleteGammaFunction.
html.[8] Macdonald Function, Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Macdonald_

function.[9] Wolfram Research, Bessel Functions. http://functions.wolfram.com/PDF/Bessel/03.04.21.0014.01.[10] Wikipedia, Meijer G Function, (2025). https://en.wikipedia.org/wiki/Meijer/G-function.[11] Wolfram Research, Meijer G. https://functions.wolfram.com/PDF/MeijerG.pdf.[12] Wolfram Research Inc. Mathematica Version 13.0, (2022). https://www.wolfram.com/mathematica/new-in-13.

https://doi.org/10.28924/ada/ma.5.19
https://lib.ugent.be/catalog/ebk01:1000000000789815
https://lib.ugent.be/catalog/ebk01:1000000000789815
https://mathworld.wolfram.com/IncompleteGammaFunction.html
https://mathworld.wolfram.com/IncompleteGammaFunction.html
https://encyclopediaofmath.org/wiki/Macdonald_function
https://encyclopediaofmath.org/wiki/Macdonald_function
http://functions.wolfram.com/PDF/Bessel/03.04.21.0014.01
https://en.wikipedia.org/wiki/Meijer/G-function
https://functions.wolfram.com/PDF/MeijerG.pdf
https://www.wolfram.com/mathematica/new-in-13

	1. Introduction
	2. Density Functions
	3. Cumulative Distribution Functions
	4. Percentiles
	References

