linear combination, product and ratio of normal ... - Semantic Scholar
K Y B E R N E T I K A — V O L U M E 41 ( 2 0 0 5 ) , N U M B E R 6, P A G E S
787-798
LINEAR COMBINATION, PRODUCT AND RATIO OF NORMAL AND LOGISTIC RANDOM VARIABLES SARALEES NADARAJAH
The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this note, the exact distributions of aX+f3Y, \XY\ and | K / ^ | are derived when X and Y are independent normal and logistic random variables. The normal and logistic distributions have been two of the most popular models for measurement errors in engineering. Keywords: linear combination of random variables, logistic distribution, normal distribu tion, products of random variables, ratios of random variables AMS Subject Classification: 62E15 1. I N T R O D U C T I O N The distributions of linear combinations, products and ratios of r a n d o m variables arise in m a n y areas of engineering. In this note, we study t h e exact distributions of aX + (3Y, \XY\ and | . X / y | when X and Y are independent r a n d o m variables having t h e normal and logistic distributions with pdfs
and
My)
_ 'exp(_^){1+exp(_^)}-2,
(2)
respectively, for —00 < x < 00, —00 < y < 00, —00 < [i < 00, —00 < A < 00, a > 0 and 0. We assume without loss of generality t h a t a > 0. Note t h a t t h e n o r m a l and logistic distributions are two of t h e most popular models for measurement errors in engineering. The calculations of this note involve several special functions, including t h e com plementary error function defined by erfc(x)
=
2 f°° -7= / VŤr Jx
exp(-t2)dt
788
S. NADARAJAH
and the hypergeometric function defined by
G(ol6c, 0, /•CO
xn exp(—px)eifc(cx + b) dx
/
- < : >4^^H»+£)}L e m m a 2.
(Equation (2.8.5.14), Prudnikov et al. [2, Vol.2]) For p > 0, /•OO
/
.
xa~l exp (—p/x) erfc(co;) dx
p °r(-«)-^rV( - a -i) G (i;i^,i + 2. where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
(6) can be expanded as
™-if-(-5){'-> (-i)r*(^^^)2 ^/:-(i){—(i)}" K^^^)^/>(^^^)g(?)-{^}-
=i£a 2 )fe X p{-^}K^^^)+
2
ga )/:-{^}*(^^)-
^ fc=0
Using the relationship
Ф(-x) =
r r f c Ш'
790
S. NADARAJAH
(7) can be further rewritten as
™ - sS^r-f^J-C^^r^)2
4g(-, )/:-{-^} The two integrals in (8) can be calculated by direct application of Lemma 1. The result follows. • The following corollaries provide the cdfs for the sum and the difference of the normal and logistic random variables. Corollary 1. Suppose X and Y are distributed according to (1) and (2), respectively. Then, the cdf of Z = X + Y can be expressed as
*w = i g s r i ^ M ^ ) - ™ - * « } • w where
*«., - e
^
+
*i±I^}e rfc (^ + x7 d
1
>\"Xч
"**•
8 . d 8_
'
-2
5
i
i
-4
•" i
-
2
^x_
i
0
2
i
4
Fig. 2. Plots of the pdf of (3) for X/ = 0.2,1,2,3, n/o = 0, /a = 1, <j) = 1, and (a): a = 1 and (3 = 1; (b): a = 1 and (3 = —1; (c): a = 1 and (3 = 2; and, (d): a = 1 and (3 = —2. The curves with the lowest to the highest modality correspond to increasing values of X/cj>.
Theorem 2. Suppose X and Y are distributed according to (1) and (2), respec tively. Then, the cdf of Z = \XY\ can be expressed as F(z)
=
3 . 1
A2(fc + l ) 2 z 2 \
"í=žUJ\^ V2 2 '2
8^—)
Kz
~ /-2\Í3O_/1
G
\(k + l)z f +_ G 1,2
^7" l
; ,1
;
3 3, A2(fc + l ) 2 z 2 \ l
'2'2'
8^ Jj'
where C denotes Euler's constant. P r o o f . The cdf F(z) = Pi(\XY\
< z) can be expressed as
(11)
Linear Combination, Product and Ratio of Normal and Logistic Random Variables
(a)
793
(b) -íd
co d •
r^
J
u.
.•••.ч
*"•"
Q û.
•••
cм d
/*• ** ~ "• •* -Л
V
'
1 1 - 4 - 2
1 0
1 2
,rf
d
ч
І
* *.
o d
i
/.••. •••."ч /. — - л
\
^1
4
1 1 - 4 - 2
% ;
- * * . * • : ' • '
1 0
"
1 2
""•"•* *.C
1 4
«*)
(c)
Fig. 3. Plots of the pdf of (3) for X/<j) = 0, \i/o = 0, <j)/a = 0.2,0.5,1,3, = 1, and (a): a = 1 and (5 = 1; (b): a = 1 and (3 = - 1 ; (c): a = 1 and /? = 2; and, (d): a = 1 and /3 = — 2. The curves from the bottom to the top correspond to increasing values of <j>/cr.
Loo
exp(-Лi/) õdy — 1, {l + H y | / { i + exp(-Лy)} 2
(12)
where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
o+-)- -
CO
Ľ(;У-
(6) can be expanded as
*'> -
2A
Ф
exp(-Лy)
Г Ш ( Г{lT+ exp(-Лy)}õdy 2
^•ШÏÏÄ*6
1
1
- f Ч^Ш *^ ^ 2A
794
S. NADARAJAH
+aA #
/l (:Bi)S(t)--«* +1)A »>*- 1
- 2 \|("* 2 )f*(^) exp{ - < * +1)A! ' ,d! ' 1
^(^/IK^H^ '^=
4xf)^
2
)
/ ~ * ^ ) e x p { - ( * + l)Ai/}dt/-l
1 (13)
Using the relationship *(-*)
=
^erfc(^),
(7) can be rewritten as ^. F(z) = 2 A _ ^ ( ~ 2 ) ^ ° ° e r f c ( - - ^ - ) e x p { - ( A ; + l ) A y } d j / - l = 2 A _ _ Y " 2 ) f°° w-2erfc(—^\exp{-(k
+ l)X/w}dw-l.
(14)
Direct application of Lemma 2 shows that the integral in (14) can be calculated as /
tu -2 erfc I
—— ) exp{—(k +
l)X/w}dw
jO
V V2(7/ 1 3C, A 3 1 A2(fc + 1 ) 2 2 2 \ + J A(fc + l) V2lFa \ 2 2 ' ' 2 5 8\ y/2z
(16)
The corresponding pdf is
___ fcГTT í , 2 Ì Ы 2 ) + 9k(-z)} , + i V fc
/(-)
(17)
fc=o
where #fc is the derivative of Gk given by 2 ' *2(fc ' + l ) V + 2A(fc + l ) ^ \ /A
A 5fe(2)
= 7^
e x p
(
V_(C7Z ЄXp
2^ \/2z + '
) A(fc + l)cг
y/2z
(18)
796
S. NADARAJAH
P r o o f . The cdf F(z) = P r ( | X / F | < z) can be expressed as
™-
чiн^нч^)}
exp(-Ay)
jdy,
{l+exp(-Ay)}2
(19)
where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
('+->- - g ( ; y . (19) can be expanded as
F{z) =
x
[1o
\rU(^±M)^(^M)\ l
+f
V
o
)
\
exp(-Ay)
2 d 2!
;/{i+exp(-Ay)}
a
U(» + z\y\\ Ť f>- 2 M)]
ex
p(Ay)
~2 d У
{l + exp(Ay)}
=iгн^м^)}m^
k+m
ЧLH^H^ЩiУ^+^ (fc+i ШfM^)-Ч^)H- >^ + d ШШ^)-K^)Ь 0 then the cdf of Z = \X/Y\ takes the form (15), where n
, ,
Gk(z)
^A 2 (t + l ) V \
=
exp(^
^
, (X(k + l)a\
jerfc(—^—
2
j .
The corresponding pdf takes the form (17), where gk is the derivative of Gk given by 9k(z)
=
X(k + 1)0-
(X2(k + lýa2\
6XP
jŠ*~
\
2z
2
K
) ^
C X P
(
{
A2(fc + l ) V \
2^—)
(*$*)}•
-y/ŤžX(k + l)crerfc
P r o o f . The proof follows by limiting /i/a —> 0 in (16).
q
09
ö
787-798
LINEAR COMBINATION, PRODUCT AND RATIO OF NORMAL AND LOGISTIC RANDOM VARIABLES SARALEES NADARAJAH
The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this note, the exact distributions of aX+f3Y, \XY\ and | K / ^ | are derived when X and Y are independent normal and logistic random variables. The normal and logistic distributions have been two of the most popular models for measurement errors in engineering. Keywords: linear combination of random variables, logistic distribution, normal distribu tion, products of random variables, ratios of random variables AMS Subject Classification: 62E15 1. I N T R O D U C T I O N The distributions of linear combinations, products and ratios of r a n d o m variables arise in m a n y areas of engineering. In this note, we study t h e exact distributions of aX + (3Y, \XY\ and | . X / y | when X and Y are independent r a n d o m variables having t h e normal and logistic distributions with pdfs
and
My)
_ 'exp(_^){1+exp(_^)}-2,
(2)
respectively, for —00 < x < 00, —00 < y < 00, —00 < [i < 00, —00 < A < 00, a > 0 and 0. We assume without loss of generality t h a t a > 0. Note t h a t t h e n o r m a l and logistic distributions are two of t h e most popular models for measurement errors in engineering. The calculations of this note involve several special functions, including t h e com plementary error function defined by erfc(x)
=
2 f°° -7= / VŤr Jx
exp(-t2)dt
788
S. NADARAJAH
and the hypergeometric function defined by
G(ol6c, 0, /•CO
xn exp(—px)eifc(cx + b) dx
/
- < : >4^^H»+£)}L e m m a 2.
(Equation (2.8.5.14), Prudnikov et al. [2, Vol.2]) For p > 0, /•OO
/
.
xa~l exp (—p/x) erfc(co;) dx
p °r(-«)-^rV( - a -i) G (i;i^,i + 2. where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
(6) can be expanded as
™-if-(-5){'-> (-i)r*(^^^)2 ^/:-(i){—(i)}" K^^^)^/>(^^^)g(?)-{^}-
=i£a 2 )fe X p{-^}K^^^)+
2
ga )/:-{^}*(^^)-
^ fc=0
Using the relationship
Ф(-x) =
r r f c Ш'
790
S. NADARAJAH
(7) can be further rewritten as
™ - sS^r-f^J-C^^r^)2
4g(-, )/:-{-^} The two integrals in (8) can be calculated by direct application of Lemma 1. The result follows. • The following corollaries provide the cdfs for the sum and the difference of the normal and logistic random variables. Corollary 1. Suppose X and Y are distributed according to (1) and (2), respectively. Then, the cdf of Z = X + Y can be expressed as
*w = i g s r i ^ M ^ ) - ™ - * « } • w where
*«., - e
^
+
*i±I^}e rfc (^ + x7 d
1
>\"Xч
"**•
8 . d 8_
'
-2
5
i
i
-4
•" i
-
2
^x_
i
0
2
i
4
Fig. 2. Plots of the pdf of (3) for X/ = 0.2,1,2,3, n/o = 0, /a = 1, <j) = 1, and (a): a = 1 and (3 = 1; (b): a = 1 and (3 = —1; (c): a = 1 and (3 = 2; and, (d): a = 1 and (3 = —2. The curves with the lowest to the highest modality correspond to increasing values of X/cj>.
Theorem 2. Suppose X and Y are distributed according to (1) and (2), respec tively. Then, the cdf of Z = \XY\ can be expressed as F(z)
=
3 . 1
A2(fc + l ) 2 z 2 \
"í=žUJ\^ V2 2 '2
8^—)
Kz
~ /-2\Í3O_/1
G
\(k + l)z f +_ G 1,2
^7" l
; ,1
;
3 3, A2(fc + l ) 2 z 2 \ l
'2'2'
8^ Jj'
where C denotes Euler's constant. P r o o f . The cdf F(z) = Pi(\XY\
< z) can be expressed as
(11)
Linear Combination, Product and Ratio of Normal and Logistic Random Variables
(a)
793
(b) -íd
co d •
r^
J
u.
.•••.ч
*"•"
Q û.
•••
cм d
/*• ** ~ "• •* -Л
V
'
1 1 - 4 - 2
1 0
1 2
,rf
d
ч
І
* *.
o d
i
/.••. •••."ч /. — - л
\
^1
4
1 1 - 4 - 2
% ;
- * * . * • : ' • '
1 0
"
1 2
""•"•* *.C
1 4
«*)
(c)
Fig. 3. Plots of the pdf of (3) for X/<j) = 0, \i/o = 0, <j)/a = 0.2,0.5,1,3, = 1, and (a): a = 1 and (5 = 1; (b): a = 1 and (3 = - 1 ; (c): a = 1 and /? = 2; and, (d): a = 1 and /3 = — 2. The curves from the bottom to the top correspond to increasing values of <j>/cr.
Loo
exp(-Лi/) õdy — 1, {l + H y | / { i + exp(-Лy)} 2
(12)
where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
o+-)- -
CO
Ľ(;У-
(6) can be expanded as
*'> -
2A
Ф
exp(-Лy)
Г Ш ( Г{lT+ exp(-Лy)}õdy 2
^•ШÏÏÄ*6
1
1
- f Ч^Ш *^ ^ 2A
794
S. NADARAJAH
+aA #
/l (:Bi)S(t)--«* +1)A »>*- 1
- 2 \|("* 2 )f*(^) exp{ - < * +1)A! ' ,d! ' 1
^(^/IK^H^ '^=
4xf)^
2
)
/ ~ * ^ ) e x p { - ( * + l)Ai/}dt/-l
1 (13)
Using the relationship *(-*)
=
^erfc(^),
(7) can be rewritten as ^. F(z) = 2 A _ ^ ( ~ 2 ) ^ ° ° e r f c ( - - ^ - ) e x p { - ( A ; + l ) A y } d j / - l = 2 A _ _ Y " 2 ) f°° w-2erfc(—^\exp{-(k
+ l)X/w}dw-l.
(14)
Direct application of Lemma 2 shows that the integral in (14) can be calculated as /
tu -2 erfc I
—— ) exp{—(k +
l)X/w}dw
jO
V V2(7/ 1 3C, A 3 1 A2(fc + 1 ) 2 2 2 \ + J A(fc + l) V2lFa \ 2 2 ' ' 2 5 8\ y/2z
(16)
The corresponding pdf is
___ fcГTT í , 2 Ì Ы 2 ) + 9k(-z)} , + i V fc
/(-)
(17)
fc=o
where #fc is the derivative of Gk given by 2 ' *2(fc ' + l ) V + 2A(fc + l ) ^ \ /A
A 5fe(2)
= 7^
e x p
(
V_(C7Z ЄXp
2^ \/2z + '
) A(fc + l)cг
y/2z
(18)
796
S. NADARAJAH
P r o o f . The cdf F(z) = P r ( | X / F | < z) can be expressed as
™-
чiн^нч^)}
exp(-Ay)
jdy,
{l+exp(-Ay)}2
(19)
where $(•) denotes the cdf of the standard normal distribution. Using the series expansion
('+->- - g ( ; y . (19) can be expanded as
F{z) =
x
[1o
\rU(^±M)^(^M)\ l
+f
V
o
)
\
exp(-Ay)
2 d 2!
;/{i+exp(-Ay)}
a
U(» + z\y\\ Ť f>- 2 M)]
ex
p(Ay)
~2 d У
{l + exp(Ay)}
=iгн^м^)}m^
k+m
ЧLH^H^ЩiУ^+^ (fc+i ШfM^)-Ч^)H- >^ + d ШШ^)-K^)Ь 0 then the cdf of Z = \X/Y\ takes the form (15), where n
, ,
Gk(z)
^A 2 (t + l ) V \
=
exp(^
^
, (X(k + l)a\
jerfc(—^—
2
j .
The corresponding pdf takes the form (17), where gk is the derivative of Gk given by 9k(z)
=
X(k + 1)0-
(X2(k + lýa2\
6XP
jŠ*~
\
2z
2
K
) ^
C X P
(
{
A2(fc + l ) V \
2^—)
(*$*)}•
-y/ŤžX(k + l)crerfc
P r o o f . The proof follows by limiting /i/a —> 0 in (16).
q
09
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