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Some characteristic properties of normal distribution
Computational North-Holland
Statistics
& Data Analysis
10 (1990) 117-120
117
Some characteristic properties of normal distribution M. Ahsanullah
*
Rider College, Lawrenceville,
NJ 08648, USA
Received April 1989 Revised November 1989 Abstract: Suppose Xi, X, are two independently distributed random variables with L, = alXl + a, X2 and W = (L, - E( L1))2/Var( L,), for Var( L,) < 00. If X’s are normally distributed then L, and W are distributed respectively as normal and &i-squared with one degree of freedom. However, if W is distributed as &i-squared with one degree of freedom then L, need not be normally distributed. In this paper some characterizations of the normal distribution are given when W is distributed as &i-squared with one degree of freedom. AMS 1970 Subject Classification: Primary 62 E 10 Secondary 60 E 05. Keywords: Symmetric
Normal distribution, Identically distribution, Characterization.
distributed
random
variables,
Characteristic
function,
1. Introduction and main results The random variable (T.v.) X is said to have the normal distribution N(p, a2) with mean p and variance u2, if its probability density function (pdf) is Qx)
= (2aa2)-1’2
exp( - (x - ~)~(20’)-~),
-ccoox,j.L~cc,fJ~o. (1.1)
The r.v. V is said to be distributed &i-squared as F/z X2(n), if its pdf f2( u) is
with n degrees of freedom, written
f2 (u) = (2”/‘r( n/2)) -‘u(~/~)-~ exp( - u/2),
u 2 0.
0.2)
The normal distribution plays an important role in statistics. Many inferences are based on the property that U = X2 A X2(1) where X has the distribution N(0, 1). However if X2 2 X2(1), then X may not be distributed as N(0, 1). The following example shows that E( 2) = 0, Var( 2) = 1, 2’ A x2(1) but 2 is not distributed as N(0, 1). * The author paper.
thanks
0167-9473/90/$3.50
the referees
for helpful
0 1990 - Elsevier
suggestions
Science Publishers
which resulted
in the improvement
B.V. (North-Holland)
of the
118
M. Ahsanullah
Example 1. f(z)
/ Properties of normal distribution
Let 2 be a random variable with pdf given by
= (2&-Y = 2(277)
if Izl>l 1
2e-:r2
= 0
if -l
-aandO
a>0
otherwise,
where a is determined such that a ze-+ dz + -aZe-+ dz = 0. J0 J -1 Lemma 1. Let Z2 2 x*(l) and C#Q( t) be the characteristic function of Z, then G,(t) + &(-t) = 2e-(1/2)‘2. For proof see Ahsanullah and Hamedani (1988). The following results easily follows from the Lemma 1. Result 1. symmetric L,/,/m.
Suppose Xi and X2 are independent and identically distributed (about zero) random variables. Let L, = qX, + a2 X2 and W = If IV2 2 x2(l), then Xi and X2 are normally distributed.
The proof can be established as follows: IV2 A X2(l) and W is symmetric about zero (weighted sum of two independent symmetric (about zero) distribution is also symmetric (about zero)). Thus L, is normal and by Cramer’s (1936) decomposition theorem it follows that Xi and X2 are normal. However in the above result the condition of independence cannot be removed. Without the assumption of independent the decomposition theorem of Cramer is not necessarily true (see for details Basu and Ahsanullah (1983)). The following result is a consequence of Lemma 1. Result 2. Suppose Xi and X2 are independent and identically distributed r.v.s. and let U = min( Xi, X2), V= max( Xi, X2). If U2 2 V2 and with pdf f(x) U2 2 x2(l), then Xi and X2 are normally distributed. P[U%u2]
P[V2su2]
=P[-UlUIU]
=P[U>
= (1 - F( -u))‘-
(1 -F(u))‘,
(A)
=P[-us
=(F(u))*-(F(-u))‘.
(B)
Vsu]
-241 -P[Uru]
Since U2 2 V2 for all 24, - cc < u < cc, equating (A) and (B), we get on simplification F(u) = 1 - F( - u), for all U, - cc < u < cc. Thus Xi and X, are symmetrically distributed. Let +i( t) be the characteristic function of U, then @i(t) + &( -t)
= 2e-;“.
“,2( 1 - F( x))f( +1(t) = /__mme =
/
But
(C)
x) dx
00 _~eKirx2(1 - F( -x))f(x))
dx = /_mweC’r”2F(x)f(x)
dx.
M. Ahsanullah
119
/ Properties of normal distribution
Thus
-“x2(l = 2
2e-f*‘=
$,( -t)
- F( x))f( x) dx + /ca e-IfX2F( x)f( x) dx --CL)
+ &(t)
Hence
dx.
O”ePi’“f(x) J --oo
= 2JW e”*f(x) --oo
dx
Therefore f(x) is the characteristic function of the standard normal distribution. For characterization of normal distribution using linear functions of random variables see Kagan et al. (1965), Laha (1957), and Skitovitch (1954). Theorem 1. Let XI and X, be two independent and identically distributed absolutely continuous random variables and suppose L, = aXI + (1 - a2)‘j2X2, 0 -C 1a 1 < 1. If L: and X: are each distributed as chi-squared with one degree of freedom, then X, and X2 are each normally distributed. Proof.
If q(t)
is the characteristic
2e-;“=+(at)$((l 2e-;‘*=+(t)
-a*)“*t) +4(-t).
2e-;n21= _ -#(at) 2e-f(l-a’,t*
function (cf) of X, then +#(-at)J,(-(1
-a2)1’2t),
and
0.5)
Now
+$(-at),
=~((1-a2)“2t)+~(-(1-a2)1’2t),
and
4e~i”=(~(at)+~(-at))(((l-a2)1’2tj+~(-(l-a2)1’2tj). We get on simplification 2e-trz=~(at)~(-(a-a2)1’2t)+~(-at)~((l-a2)1’2t). Thus combining this with (1.5), we have (~(at)-~(-at))(~((1-a2)1’2tj-~(-(l-a2)1’2tj)=0 for all real t. Therefore \c/( t) = $J( - t) for all real t and $(at)$((l Hence q(t)
- a*)“*t)
= e-;“.
is the cf of the standard normal distribution.
References [l] Ahsanullah, M. (1987). A note on the characterization of the normal distribution. Biom. J. 29, 7, 885-888.
120
M. Ahsanullah
/ Properties of normal distribution
[2] Ahsanullah, M. and Hamedani, G.G. (1988) Some characterizations of the normal distribution. Cal. Stat. Assoc. Bull. 37, 95-99. [3] Basu, A.K. and Ahsanullah, M. (1983). On a decomposition of normal and Poisson law. Cal. Stat. Assoc. Bul, 32, 103-109. [4] Cramer, H. (1936). Uber eine eigenschaft der normalen verteilungs function. Math. Zeitschrift 41, 405-411. [5] Laha, R.G. (1957). On a characterization of the normal law from properties of suitable linear statistics. Ann. Math. Statist. 28, 126-139. [6] Kagan A.M., Linnik, Yu and Rao, E.R. (1965). On a characterization of the normal law based on a property of the sample average. Sankhya A 27, 405-406. [7] Linnik, Yu. V. and Zinger, A.A. (1955). On a analytic generalization of the Cramer theorem (in Russian). Vestnik Leningrad Univer. 11, 51-56. [8] Rao, C.R. (1967). On some characterizations of the normal law. Sankhya A 29, 1-14. [9] Skitovitch, V.P. (1954). Linear forms of independent random variables and the normal distribution law (in Russian). Izvestiya Acad. Nauk SSSR Ser. Matem. 18, 185-200.
Statistics
& Data Analysis
10 (1990) 117-120
117
Some characteristic properties of normal distribution M. Ahsanullah
*
Rider College, Lawrenceville,
NJ 08648, USA
Received April 1989 Revised November 1989 Abstract: Suppose Xi, X, are two independently distributed random variables with L, = alXl + a, X2 and W = (L, - E( L1))2/Var( L,), for Var( L,) < 00. If X’s are normally distributed then L, and W are distributed respectively as normal and &i-squared with one degree of freedom. However, if W is distributed as &i-squared with one degree of freedom then L, need not be normally distributed. In this paper some characterizations of the normal distribution are given when W is distributed as &i-squared with one degree of freedom. AMS 1970 Subject Classification: Primary 62 E 10 Secondary 60 E 05. Keywords: Symmetric
Normal distribution, Identically distribution, Characterization.
distributed
random
variables,
Characteristic
function,
1. Introduction and main results The random variable (T.v.) X is said to have the normal distribution N(p, a2) with mean p and variance u2, if its probability density function (pdf) is Qx)
= (2aa2)-1’2
exp( - (x - ~)~(20’)-~),
-ccoox,j.L~cc,fJ~o. (1.1)
The r.v. V is said to be distributed &i-squared as F/z X2(n), if its pdf f2( u) is
with n degrees of freedom, written
f2 (u) = (2”/‘r( n/2)) -‘u(~/~)-~ exp( - u/2),
u 2 0.
0.2)
The normal distribution plays an important role in statistics. Many inferences are based on the property that U = X2 A X2(1) where X has the distribution N(0, 1). However if X2 2 X2(1), then X may not be distributed as N(0, 1). The following example shows that E( 2) = 0, Var( 2) = 1, 2’ A x2(1) but 2 is not distributed as N(0, 1). * The author paper.
thanks
0167-9473/90/$3.50
the referees
for helpful
0 1990 - Elsevier
suggestions
Science Publishers
which resulted
in the improvement
B.V. (North-Holland)
of the
118
M. Ahsanullah
Example 1. f(z)
/ Properties of normal distribution
Let 2 be a random variable with pdf given by
= (2&-Y = 2(277)
if Izl>l 1
2e-:r2
= 0
if -l
-aandO
a>0
otherwise,
where a is determined such that a ze-+ dz + -aZe-+ dz = 0. J0 J -1 Lemma 1. Let Z2 2 x*(l) and C#Q( t) be the characteristic function of Z, then G,(t) + &(-t) = 2e-(1/2)‘2. For proof see Ahsanullah and Hamedani (1988). The following results easily follows from the Lemma 1. Result 1. symmetric L,/,/m.
Suppose Xi and X2 are independent and identically distributed (about zero) random variables. Let L, = qX, + a2 X2 and W = If IV2 2 x2(l), then Xi and X2 are normally distributed.
The proof can be established as follows: IV2 A X2(l) and W is symmetric about zero (weighted sum of two independent symmetric (about zero) distribution is also symmetric (about zero)). Thus L, is normal and by Cramer’s (1936) decomposition theorem it follows that Xi and X2 are normal. However in the above result the condition of independence cannot be removed. Without the assumption of independent the decomposition theorem of Cramer is not necessarily true (see for details Basu and Ahsanullah (1983)). The following result is a consequence of Lemma 1. Result 2. Suppose Xi and X2 are independent and identically distributed r.v.s. and let U = min( Xi, X2), V= max( Xi, X2). If U2 2 V2 and with pdf f(x) U2 2 x2(l), then Xi and X2 are normally distributed. P[U%u2]
P[V2su2]
=P[-UlUIU]
=P[U>
= (1 - F( -u))‘-
(1 -F(u))‘,
(A)
=P[-us
=(F(u))*-(F(-u))‘.
(B)
Vsu]
-241 -P[Uru]
Since U2 2 V2 for all 24, - cc < u < cc, equating (A) and (B), we get on simplification F(u) = 1 - F( - u), for all U, - cc < u < cc. Thus Xi and X, are symmetrically distributed. Let +i( t) be the characteristic function of U, then @i(t) + &( -t)
= 2e-;“.
“,2( 1 - F( x))f( +1(t) = /__mme =
/
But
(C)
x) dx
00 _~eKirx2(1 - F( -x))f(x))
dx = /_mweC’r”2F(x)f(x)
dx.
M. Ahsanullah
119
/ Properties of normal distribution
Thus
-“x2(l = 2
2e-f*‘=
$,( -t)
- F( x))f( x) dx + /ca e-IfX2F( x)f( x) dx --CL)
+ &(t)
Hence
dx.
O”ePi’“f(x) J --oo
= 2JW e”*f(x) --oo
dx
Therefore f(x) is the characteristic function of the standard normal distribution. For characterization of normal distribution using linear functions of random variables see Kagan et al. (1965), Laha (1957), and Skitovitch (1954). Theorem 1. Let XI and X, be two independent and identically distributed absolutely continuous random variables and suppose L, = aXI + (1 - a2)‘j2X2, 0 -C 1a 1 < 1. If L: and X: are each distributed as chi-squared with one degree of freedom, then X, and X2 are each normally distributed. Proof.
If q(t)
is the characteristic
2e-;“=+(at)$((l 2e-;‘*=+(t)
-a*)“*t) +4(-t).
2e-;n21= _ -#(at) 2e-f(l-a’,t*
function (cf) of X, then +#(-at)J,(-(1
-a2)1’2t),
and
0.5)
Now
+$(-at),
=~((1-a2)“2t)+~(-(1-a2)1’2t),
and
4e~i”=(~(at)+~(-at))(((l-a2)1’2tj+~(-(l-a2)1’2tj). We get on simplification 2e-trz=~(at)~(-(a-a2)1’2t)+~(-at)~((l-a2)1’2t). Thus combining this with (1.5), we have (~(at)-~(-at))(~((1-a2)1’2tj-~(-(l-a2)1’2tj)=0 for all real t. Therefore \c/( t) = $J( - t) for all real t and $(at)$((l Hence q(t)
- a*)“*t)
= e-;“.
is the cf of the standard normal distribution.
References [l] Ahsanullah, M. (1987). A note on the characterization of the normal distribution. Biom. J. 29, 7, 885-888.
120
M. Ahsanullah
/ Properties of normal distribution
[2] Ahsanullah, M. and Hamedani, G.G. (1988) Some characterizations of the normal distribution. Cal. Stat. Assoc. Bull. 37, 95-99. [3] Basu, A.K. and Ahsanullah, M. (1983). On a decomposition of normal and Poisson law. Cal. Stat. Assoc. Bul, 32, 103-109. [4] Cramer, H. (1936). Uber eine eigenschaft der normalen verteilungs function. Math. Zeitschrift 41, 405-411. [5] Laha, R.G. (1957). On a characterization of the normal law from properties of suitable linear statistics. Ann. Math. Statist. 28, 126-139. [6] Kagan A.M., Linnik, Yu and Rao, E.R. (1965). On a characterization of the normal law based on a property of the sample average. Sankhya A 27, 405-406. [7] Linnik, Yu. V. and Zinger, A.A. (1955). On a analytic generalization of the Cramer theorem (in Russian). Vestnik Leningrad Univer. 11, 51-56. [8] Rao, C.R. (1967). On some characterizations of the normal law. Sankhya A 29, 1-14. [9] Skitovitch, V.P. (1954). Linear forms of independent random variables and the normal distribution law (in Russian). Izvestiya Acad. Nauk SSSR Ser. Matem. 18, 185-200.