A remark on the distribution of products of independent normal random

− 2 f

i

(t) :=

σ √

2π exp

ξ2

n 2

1 n

2 1

f (t) :=

σ √

exp 2π −

2

normal random variables

Jerzy Szczepański

^*

a Jagiellonian Universtity, Faculty of Mathematics and Computer Science, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland

Article history Received: 6 January 2021 Received in revised form:

8 March 2021

Accepted: 9 March 2021 Available: 10 March 2021

Abstract

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

Keywords: normal distribution, Meijer G-functions

Introduction

Basic statistic tests are constructed with the aim of sums ξ

1

+ ξ

2

+ · ·· + ξ

n

, squares ξ

²

, ξ

²

, ... , ξ

²

, sum of squares ξ

²

+ ξ

²

+ · ·· + ξ

²

, quotients

^ξ1

or other algebraic operations on sequences of independent random variables ξ

1

, ξ

2

, ..., ξ

n

(see eg. [6]).

While the sum ξ

1

+ ξ

2

+ · ·· + ξ

n

has a density that is commonly easy to determine other algebraic operations lead to densities that are more and more complicated and demand special functions in analysis. For example the sum of the independent identically distributed (abbreviated as i.i.d.) random variables ξ

i

, i = 1, 2,. .., n with the normal N (m

i

, σ

i

) probability density function (p.d.f. for abbreviation)

1 (

1 ( t − m

i 2

has p.d.f. of the same family of normal distributions N (m, σ) 1 (

1 ( t − m

2

σ

* Corresponding author: Jer zy.Szczepanski@uj.edu.pl

ISSN 2544-9125 Science, Technology and Innovation, 2020, 10 (3), 30–37

© 2021 University of Applied Sciences in Tarnow. Published under the Creative Commons Attribution 4.0 (CC BY-NC) International License

www.stijournal.pl i

(1)

i

σ

2 Meijer G-functions and their basic proper-

ties

n

0,m

p,q

2

0,2

IT II

1 2 n

1 2 n

n n ^{2 2}

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

with m := m

1

+ m

2

+ · ·· + m

n

and σ

²

= σ

²

+ σ

²

+ · ·· + σ

²

. The sum ξ

²

+ ξ

²

+ · ·· + ξ

²

of i.i.d. random variables with standard normal p.d.f. (i.e. N (m, σ) with m = 0 and σ = 1) is the χ

²

distribution with n degrees of freedom, i.e. has p.d.f.

defined by

 1

t

^{n −1}

e

^−t

, t ≥ 0 f (t) := 2

²

Γ(

₂

)

0, t < 0

which requires the use of the Euler gamma function Γ. It is known (see [9], [10]) that the answer to a very simple question about the distribution of the product of normally distributed random variables requires the use of the Meijer G-functions G

^m,0

, see (2). We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables in terms of Meijer G functions using the multiplicative convolution formula for Meijer G-functions (8). In some special cases the function G

^m,n

can be reduced to a simpler form (see [1]), for example standard normal distribution N (0, 1) has the p.d.f. given by

1 t

1

1,0

t

²

− − f (t) = √

2π exp −

2 = √

G

_0,1

2π 2 0 −

The function G

^2,0

(x) can be expressed with the modified Bessel function of the second kind K

ν

, see (11).

Meijer G-functions and their basic properties

The G- function was introduced by C.S. Meijer in 1936 and is defined in terms of Mellin-Barnes type integrals (see [1], [2] and references given there)

m,n

a 1 /

z =

m

Γ(b

k

−

k=1

ζ)

q

n

Γ(1 − a

k

+ ζ)

k=1 ζ

p

G

_p,q

b 2πi

_k=

IT

m+1

Γ(1 − b

k

+ ζ)

k=

II

n+1

Γ(a

k

− ζ) z dζ (2)



L

Sci, Tech Innov, 2020, 10 (3), 30–37

l.pl

where a = (a

1

,. .., a

n

, a

n+1

,. .., a

p

) and b = (b

1

,. .., b

m

, b

m+1

,. .., b

q

)

are se- quences of real or complex parameters, with the contour of integration L

suitably chosen on the complex plane C. If m = 0, n = 0, p = 0 or q = 0 we put 1

in place of the product over an empty set of indices and we put a sign of

omission ‘−’ in

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

r

ζ

b

¹

, b

²

,. . . , b

m

G

^m,0

(z|b

1

, b

2

, . . . , b

m

) = G

^m,0

( z1 − −

z

^σ

G

^m,n

(

z 1a

¹

, . . . , a

p

= G

^m,n

(

z 1a

¹

+ σ, . . . , a

p

+ σ

I

∞

G

^m,n

(

ηx 1a

¹

, . . . , a

p

G

^µ,ν

(

ωx 1c

¹

, . . . , c

σ

dx

( ω 1−b

1

,. . . ,

−b

_∞

x^α−1G^m,n (

ηx1a1, . . . , ap

) G^µ,ν

(

ωx1c1, . . . , cσ

)

dx

= η^−αG^n+µ,m+ν

( ω 1−b¹ − α + 1, . . . , −bm − α + 1, c1, . . . , cσ, −bm+1 − α + 1, . . . , −bq − α + 1)

_∞

x^α−1G^m,n (

ηx1a1, . . . , ap

) G^µ,ν

( 1 1c¹, . . . , cσ

)

dx

= _∞

x^α−1G^m,n (

ηx1a1, . . . , ap

) G^ν,µ

(

ωx11 − d1, . . . , 1 − dτ

)

dx

= η^−αG^n+ν,m+µ

( ω 1−b¹ − α + 1, . . . , −bm − α + 1, 1 − d1, . . . , 1 − dτ , −bm+1 − α + 1, . . . , −bq − α + 1)

ω 1b1 + α,... , bm + α, d1,..., dτ , bm+1 + α,..., bq + α

an appropriate place in the symbol of the G-function or omitting the sign ‘−’ if this does not lead to confusion. For example

= 1 2πi Γ(b

1 L

− ζ)Γ(b

2

− ζ) . ..

Γ(b

m

− ζ)z dζ

We recall selected basic formulae involving Meijer G-functions (see [1], [2], [7]). By the definition of G-function we have

p,q

( 1 b

1

,. . . , b

q p,q

1 b

1

+ σ,.. ., b

q

+ σ 1a ,. .., a ( 1 1 − b ,. .., 1

− b

(3) G

_p,q^m,n

z

⁻¹

1

^{1 p}

= G

^n,m

z1

^{1 q}

.

1 b

1

,. . . , b

q q,p

11 − a

1

,. . . , 1 − a

p

The classical integral formula for Meijer G-functions is of fundamental importance in calculations:

p,q 0

1

1 b

1

,. . . , b

q σ,τ

1d

1

,. . . , d

τ

m

, c

¹

,. .., c

^σ

,

−b

m+1

,. .., −b

^q

= η G

_q+σ,p+τ

η 1−a

1

,. .., −a

n

, d

1

,. .., d

τ

, −a

n+1

,. .., −a

p

By (3) and (4) we get a slightly more general

0 p,q 1 b1,... , bq σ,

τ 1d1,... , dτ

= η^1−α _∞

G^m,n (

ηx1a1 + α − 1, . . . , ap + α − 1) G^µ,ν

(

ωx1c1, . . . , cσ

)

dx (5)

p,q

0 1 b1 + α − 1,... , bq + α

− 1

σ,

τ 1d1,... , dτ

q+σ,p+τ

η 1−a1 − α + 1, . . . , −an − α + 1, d1, . . . , dτ , −an+1 − α + 1, . . . , −ap − α + 1

By (3) and (5) we have

p,q

0 1 b1,... , bq σ,

τ ωx 1d1,..., dτ

0 p,q 1 b1,... , bq τ,

σ 11 − c1,... , 1 − cσ q+τ,p+σ

p+σ,q+

η 1−a1 − α + 1, . . . , −an − α + 1, 1 − c1, . . . , 1 − cσ, −an+1 − α + 1, . . . , −ap − α + 1

= η^−αG^m+µ,n+ν

( η 1a¹ + α,... , an + α, c1,... , cσ, an+1 + α,... , ap + α)

0,m 0,

m

(6) n+µ,m+

ν

(4)

Sci, Tech Innov, 2020, 10 (3), 30–37

l.pl r

∞

G

^m,n

(

ηx 1a

¹

, . . . , a

p

G

^µ,ν

( 1 1c

¹

, . . . , c

σ

dx

Putting α = 0 in (6) we get

p,q

0

1 b

¹

,. . . , b

^q

σ,

τ

ωx 1d

¹

,. . . , d

^τ

x

=G

^m+µ,n+ν

( η 1a

¹

,.

. . , a

ⁿ

, c

¹

,. . . , c

σ

, a

n+1

,. . . , a

p

(7)

p+σ,q+τ

ω 1b

¹

,. .., b

^m

, d

¹

,. .., d

^τ

, b

^m+1

,. .., b

^q

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= G

( 1 1

1

p,q σ,

τ p,

q σ,

τ

x

r

∞

G

^m,n

(

x1 a

1

, . . . , a

p

)

G

^µ,ν

( t 1c

¹

, . . . , c

σ

)dx

p+σ,q+τ

By (6) we get also

1b

1

,. .., b

m

, d

1

,. .., d

τ

, b

m+1

,. .., b

q

=2

^{α −1}

r

∞

s

^{α −1}

G

^m,n

(

ηs1 a

1

, . . . , a

p

) G

^µ,ν

( 1 1c

¹

, . . . , c

σ

) ds

For ω = t

⁻¹

and η = 1 in (7) we obtain the multiplicative convolution formula for Meijer G-functions:

{G

^m,n

∗ G

^µ,ν

}(t) := r

^∞

G

^m,n

(x)G

^µ,ν

(tx

⁻¹

) dx

m+µ,n+ν p+σ,q+τ

or more specifically

0 p,q

( 1 b

1

,. .., b

qσ,τ

x 1d

¹

,. .., d

τ

x 1 a ,. .., a ,c ,. .., c ,a

,. .., a

) (8)

=G

m+µ,n+ν

t1

^{1 n 1 σ n+1 p}

r

^{∞ α−1} _m,n

( ηx

²

1a

¹

, . . . ,

a

p

)

^µ,ν

( 2 1c

¹

,. .., c

σ

) d x

G

_p,q

0

2 1 b

¹

,. ..,

b

q

G

_σ,

τ

ωx

²

1d

1

,. .., d

τ

2

2

0

2 2

p,q

1 b

1

,.

.., b

q

σ,

τ

ωs 1d

¹

,. .., d

τ

)

=2

^{α −1}

η

^−α

G

^m+µ,n+ν

0

ω b

¹

+ α,.. ., b

m

+ α, d

1

,. .., d

τ

, b

m+1

+ α,..

., b

q

+ α

(9

Sci, Tech Innov, 2020, 10 (3), 30–37

www.stijourna 1

p+σ,q+

η a

1

+ α,.. ., a

n

+ α, c

1

,. .., c

σ

, a

n+1

+ α,.. ., a

p

+ α Putting α = 0, η = 1 and ω = t

⁻¹

in (9) we get

I

^∞ α−1 m,n

( x

²

1a

¹

, . . . , a

p

)

^µ,ν

( 2t 1c

¹

,. .., c

σ

) d x G

_p,q

0

2 1 b

¹

,. .., b

q

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

G

_σ,τ

x

²

1d

1

,. .., d

τ

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l.pl

0,m

1 z

I T II

I I

II

l r

p

z

= 1

G

^m+µ,n+ν

(

t1 a

1

, .

. . , a

ⁿ

, c

1

,. .., c

σ

, a

n+1

,. .., a

p

) (10)

2

^p+σ,q+τ

1b

1

,. .., b

m

, d

1

,. .., d

τ

, b

m+1

,. .., b

q

We recall a formula for the G

^2,0

-function G

^2,0

(x|a, b) = G

^2,0

(x1 −−

) = 2x

^{a + b}

K

(2 √

x) (11)

0,2 0,2

1a, b

−

a−b

(see 5.6(4) in [1]), where K

ν

is the modified Bessel function of the second kind of order ν. Many other formulae to express the Meijer G-function in other special functions can be found in [1]. Most likely, we should not expect,

unfortunately, a reasonable reduction of the Meijer function G

^m,0

to simpler special functions when m > 2. In [3] some applications of the Fox H function

p,qm,n

(a

1

, α

1

),. . ., (a

p

, α

p

)

1 (b

1

, β

1

),. .., (b

q

, β

q

)

:= 1

2π i

m

Γ(b

k

− β

k

ζ)

k=1 L q

Γ(1 − b

k

+ β

k

ζ)

k=m+1

n

Γ(1 − a

k

+ α

k

ζ)

k=1 ζ

dζ

Γ(a

k

− α

k

ζ)

k=n+1

(12)

0,2

H

2

I

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

3 Main result

−

I

⁻

(t) =

f (tx

r= f

(x)f

ξ1ξ2

π

⁰

ξ

ξ1 ξ1

were investigated to study the distribution of products, quotients and powers of independent random variables. However the Fox H function generalizes the Meijer G function and we should not expect all the more any simplification of results formulated in terms of the Fox H function to more elementary functions.

Main result

Theorem 3.1 Let ξ

1

, ξ

2

,. .., ξ

m

be i.i.d. random variables with standard normal p.d.f. with m

i

= 0 and σ

i

= 1, see (1). Then the product ξ

1

ξ

2

.. . ξ

m

has p.d.f. given by

f

_ξ1ξ2...ξm

(t) = (2π)

^−m/2

G

^m,0

(2

^−m

t

²

| 0, 0,. .., 0 ) 1

_m,00,

(

t

2 m

-.

par

..

a

,.

meter s

= (13) ( √

2π)

^m

G

_0,m

In the case when m = 2 we have

2

^m

0, 0,. .., 0 −

f (t) = 1

K (|t|) (14)

where K

0

is the modified Bessel function of the second kind of order 0.

Proof. By the formula for the p.d.f. of the product ξη of two independent random variables ξ and η with p.d.f. f

ξ

and f

η

, respectively, we get (see eg.

[6]):

f

_ξη ^∞

dx

ξ 1

η

(x)

−∞

|x|

∞

(tx

⁻¹

) dx

−∞

|x|

Hence for two i.i.d. random variables ξ

1

, ξ

2

with standard normal p.d.f. we have

f

_ξ

(t) =

_1ξ2

r

^∞

f (tx

⁻¹

)f (x) dx

= 2 I

^∞

f (tx

⁻¹

)f (x) dx

ξ ξ

2 2

−∞

|x|

0

x

because the function f

ξ₂

is even. The last integral is the multiplicative con- volution of functions f

ξ1

and f

ξ2

:

∞

f

ξ₁

0

(tx

⁻¹

)f

ξ

dx

(x) x =: {f

ξ₁

· f

ξ

2

}(t).

r

2

Sci, Tech Innov, 2020, 10 (3), 30–37

www.stijourna r

/

p,q p^l,q

l p+p^l,q+q

l

2π 2 2

π

0,1 2

0

2π 0,1 2

=

( √ 2π)

²

G

_0,1

2 x

²

0

−

G

_0,1

2 0

G

_0,1

G

_0,1

−

ξ

By the convolution formula for Meijer G-functions, see (8):

G

^m,n

∗ G

^ml,nl

= G

^m+ml,n+nl

for

f (t) =

f (u) =

^√1

exp(−

¹

u

²

) =

^√1

G

^1,0

(

_u

2

− −)

=

^√1

G

^1,0

(

^u2

|0) and using formulae (10) and (11) we get:

f

_ξ1ξ2

(t) = 2

^∞

f (tx

⁻¹

)f

0 ξ

( x) x dx 2 r

∞

0

1,0

1 t

²

− −

_1,0

x

²

− − dx x 2 r

∞

0 1,0

t

²

ξ ξ

=

( √ 2π)

²

2

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x

²

− −

^1,0

x

²

− − dx x

2

1

^2,0

t

²

− − 1

_{2,0 −2 2}

1

= ( √ 2π)

²

·

2 G

_0,2

4 0, 0 − = 2π

G

_0,2

(2

t |0, 0) =

π K

0

(|t|)

42

Science, Technology and Innovation Original Article

Sci, Tech Innov, 2020, 10 (3), 30–37

www.stijourna

− −

Assume for m ≥ 2 the product ξ

1

ξ

2

.. . ξ

m−1

has p.d.f. of the form 1

_m−1,0

t

²

− −

− f

_{ξ1ξ2...ξm−1}

(t) =

( √

2π)

^m−1

G 2

^m−1

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r

/

=

( √ 2π)

^m

G

_0,m−

1

G

_0,1

2

^m

2 0,. .., 0 − G

_0,1

2 0

−

0,. .., 0

^m−

"

1 param eters

Proceeding as above for two factors f

ξ1

and f

ξ2

we have by (10):

f

_{ξ1ξ2...ξm−1ξm}

(t) = 2

^∞

f

₀ ^ξ2ξ2...ξm

(tx

⁻¹

)f

ξ

dx ( x) x 2 r

∞

0

m−1,0

1

2

− −

1,0

x

²

− − dx x x

t

1

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www.stijourna 2 I

∞

0 m−1,0

t

^{2 2}

− −

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

"- ..,.

_

2 x 0,. .., 0 − 2 0 G

_0,m−

−

1

1,0

x

²

− − dx x

2 1

2

− −

m,0

−

= ( √

2π)

^m

·

2 G

_0,m

=

( √

2π)

^m

Sci, Tech Innov, 2020, 10 (3), 30–37

l.pl

□ t

0,m

0,. .., 0, 0 2

^m m parameters

1

= ( √ 2π)

^m

G

^m,0

(2

^−m

t

²

|0, 0,. .., 0).

Corrolary 3.2 Let ξ

¹

, ξ

2

,. .., ξ

m

be independent normally distributed ran-

dom variables with mean values m

i

= 0 and standard deviations σ

i

> 0, i.e.

f

ξ_i

(t) = σ √ exp 2π

− σ

²

l.pl

Sci, Tech Innov, 2020, 10 (3), 30–37

with p.d.f.

1 (

i

t

²

l The p.d.f. of the product ξ

1

ξ

2

.. . ξ

m

equals

1

_m,0

(

_{−m 2}

f

ξ1ξ2...ξm

(t) = σ( √

2π)

^m

G

_0,m

2 (t/σ) |0,. .., 0

i

, i = 1, 2,. ..,

m.

Sci, Tech Innov, 2020, 10 (3), 30–37

l.pl

0,m

≥

0,2 4 0

where σ := σ

1

σ

2

.. . σ

m

. (15)

Remark 3.3 It was noticed in [4], [5] that the density (14) of the product ξ

1

ξ

2

was found in 1932 in [11]. The formula (14) is also an exercise to the reader in [8]. The density (15) of the product ξ

1

ξ

2

.. . ξ

m

was found in [9] in 1966. See also [10].

Question 3.4 Is it possible to simplify the Meijer function G

^m,0

(x|0,.

.., 0) for m > 2 as in the case of m = 2 when G

^2,0

(

^t2

|0, 0) = 2K (|t|)?

Question 3.5 Is it possible to give an explicit formula for the p.d.f. of the product ξ

1

ξ

2

.. . ξ

m

, m 2, of normally distributed independent random variables with different mean values?

We may find a partial answer to the question in [8] in a special case of the product ξ

1

ξ

2

where ξ

1

∼ N (0, σ) and ξ

2

∼ N (m, σ).

Acknowledgements

The author would like to thank the reviewers for drawing at- tention to the results published in [3], [4], [5], [8], [9], [10], [11] and for indicating an error in the formula (5) in the earlier version of the paper.

References

1. Bateman H, Erdélyi A. Higher transcendental functions.

Vol. 1. New York: McGraw-Hill; 1953.

2. Beals R, Szmigielski J. Meijer G-functions: a gentle intro- duction. Notices of the American Mathematical Society.

2013;60(7):866–872. doi: http://dx.doi.org/10.1090/

notimanid1016.

3. Carter BD, Springer MD. The distribution of products, quotients and powers of independent H-function variates. SIAM Journal on Applied Mathematics. 1977;33(4):542– 558. doi:

https://doi.org/10.1137/0133036.

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Figure 1: Plots of the p.d.f. of the products ξ

1

.. . ξ

m

for m = 1 (red line),

m = 2 (green line), m = 3 (blue line) and m = 4 (black line).

Sci, Tech Innov, 2020, 10 (3), 30–37

l.pl

Figure 2: Plots of the p.d.f. of the products ξ

1

.. . ξ

m

for m = 1 (red line),

m = 2 (green line), m = 3 (blue line) and m = 4 (black line).