ON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS

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1 Pa J Statist 2009 Vol 25(2), ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad, Paistan masoodanwar@comsatsedup 2 National College of Business dministration & Economics, Lahore, Paistan drmunir@brainnetp BSTRCT Negative moments of certain discrete probability distributions in terms of hypergeometric power series functions are obtained KEY WORDS Negative moments; discrete distributions 1 INTRODUCTION Recently negative moments have been studied by Roohi (2003) who obtained negative moments of some discrete distributions in terms of hypergeometric series functions In is paper we have extended her wor by considering furer discrete probability distributions and expressed e moments in terms of newly defined generalized hypergeometric series function Theorem 21 2 NEGTIVE MOMENTS OF SOME DISCRETE DISTRIBUTIONS Let X be a geometric-compound random variable, wi parameters α and β having probability mass function (pmf) ( ) ( x 1) ( 1) Γα Γβ Γ( α β x) Γ αβ Γ α Γ β = The negative moment of where 0 order is given by ( 1) ( αβ) 135, α > 0, β> 0, x = 1, 2, (21) β E( X ) = 3H2 ( 1, ), α,1; ( 2, ), ( αβ 1 );1, (22) Since X is a geometric-compound random variable wi parameters α and β en

2 136 On negative moments of certain discrete distributions where ( ) ( 1) ΓαβΓβ 1 Γα ( x 1) EX ( ) =, ΓαΓβ x= 1 ( x ) Γ( α β x) β ( 1) α 1 = 1 ( 1) ( αβ ) ( 2) ( αβ 1) ( 1) ( 2) α( α 1) 12 1 ( 2) ( 3) ( αβ 1)( αβ 2) 2! β = 3H2 [( 1, ), α,1;( 2, ),( αβ 1);1 ] ( 1) ( αβ) ( 1 ) ( 2 ) ( ) ( 1 )( 2 ) ( ) H a,, a,,, a, ; b, b,,, b, ; z p q p q [ 1 1 ] [ 2 2 ] [ ] [ ] 1 ( 1) ( 1) ( 1) 2 a a a a ap ap p z 1 2 q 1( 1 1) 2( 2 1) q( q 1) a a a 2 z = 1 b b b b b b b b b 2! is a generalized hypergeometric series function wi usual conditions (hmad, 2008) If = 1, en p H q = p F q If = 2, en p H q is 2 pf 2 q a 1, a 1, a 2, a 2,, ap, ap; b 1, b 1, b 2, b 2,, bq, bq; z, In general H = F p q p q If s are different say i, en If = 1, Theorem 22 en negative moment is p H q = p F q p i q i i= 1 i= 1 1 β EX ( ) = 3H2[ ( 1,1), α,1;( 2,1),( αβ 1);1] ( 1)( αβ) Let X be a beta-binomial random variable wi parameters α, α > 0, β, β> 0 and pmf n Γ( αβ) Γ ( x α) Γ ( nβ x) =, x = 0,1, 2,, n, (23) x Γα Γβ Γ ( n α β) en e negative moment of order is given by

3 Masood nwar and Munir hmad 137 EX ( ) = 3H2[ (, ), α, n;( 1, ), n β 1;1 ], > 0, (24) ΓαβΓ ( ) ( n β) where = P( X = 0) = ΓβΓ ( n α β ) Suppose X is a beta-binomial random variable wi parameters α and β en ( ) n Γαβ n Γα ( x) Γ ( nβ x) EX ( ) =, Γα ΓβΓ ( n α β ) x= 0 x ( x ) ΓαβΓ ( ) ( nβ) α( n) = 1 ΓβΓ ( n α β ) ( 1) ( n β 1) ( 1) α( α 1)( n)( n 1) 1, ( 1) ( 2) Γ( n β 1) Γ( n β 2) 2! If = 1, = 3H2 [(, ), α, n;( 1, ), n β 1;1 ] en negative moment is 1 EX ( ) = 3H2[ (,1), α, n; ( 1,1), n β 1;1 ] Theorem 23 Let X be a hypergeometric random variable, wi parameters a, a > 0, b, b > 0 and pmf a b a b =, x = 0,1,2,,min( na, ), (25) x n x n en e negative moment of order is given by EX ( ) = 3H2[ (, ), a, n;( 1, ), b n 1;1 ], > 0 (26) b!( a b n)! where = P( X = 0) = ( b n)!( a b)! Suppose X is a hypergeometric random variable wi parameters a and b en

4 138 On negative moments of certain discrete distributions n 1 a b a b E( X ) = x= 0 ( x ) x n x n b!( a b n)! ( a)( n) = 1 ( b n)!( a b)! ( 1) ( b n 1) ( 1) ( a)( a 1)( n)( n 1) 1, ( 1) ( 2) ( b n 1)( b n 2) 2! E( X ) = 3H2[ (, ), a, n;( 1, ), b n 1;1 ] If = 1, en negative moment is 1 E( X ) = 3H2[, a, n; 1, b n 1;1] Theorem 24 Let X be a Waring random variable, wi parameters a, a 2, cc, > aand pmf ( c a)( a x 1)!( c)! = ca ( 1)!( c x)! en e negative moment of order is given by, c > a 2, x = 0,1, 2, (27) EX ( ) = 3H2[ (, ), a,1;( 1, ), c 1;1 ], > 0, (28) ( c a) where = P( X = 0) = c Suppose X is a Waring random variable wi parameters a and c en c!( c a) 1 ( a x 1)! EX ( ) =, ca ( 1)! x= 0 ( x ) ( c x)! ( c a) ( a)1 ( 1) ( a)( a 1)12 1 = 1, c ( 1) ( c 1) ( 1) ( 2) ( c 1)( c 2) 2! E( X ) = 3H2[ (, ), a,1;( 1, ), c 1;1 ] If = 1, en negative moment is

5 Masood nwar and Munir hmad E( X ) = 3H2[, a,1; 1, c 1;1] Corollary 21 If a = 1, e Waring function reduces to Yule probability function and Waring results holds for Yule function Theorem 25 Let X be a random variable having Poisson-binomial distribution wi parameters n and p,0 p 1, aa>, 0 and pmf m a a nm x nm x = e p(1 p), ( nm, ) Z, x = 0,1,2,, nm (29) m= 0 m! x The negative moment of a order is given by m e a nm p E( X ) = (1 p) 2H1 (, ), nm;( 1, );, > 0, m = 0 m! 1 (210) Suppose X is a Poisson-binomial random variable and negative moment of first order is given by If = 1, nm 1 m a a nm x nm x E( X ) = e p (1 p), x= 0( x ) m= 0 m! x a m e a nm ( nm) p = (1 ) 1 p m= 0 m! ( 1) 1 2 ( 1) ( nm)( nm 1) p, ( 1) ( 2) 2! 1 p a m e a nm p EX ( ) = (1 p) 2H1 (, ), nm;( 1, ); m = 0 m! 1 en negative moment is a m 1 e a nm p EX ( ) = (1 p) 2H1 (,1), nm;( 1,1); m= 0 m! 1

6 140 On negative moments of certain discrete distributions Corollary 22 Let X be a random variable having Hermite distribution wi parameters p,0 p 1 and a, a > 0, having pmf 2m = e p(1 p) The negative moment of m a a x 2m x m= 0 m! x order is given by (210) when n = 2 REFERENCES, m Z, x = 0,1,2,,2m (211) 1 hmad, M and Saboor, (2009) Properties of a newly defined hypergeometric power series function Pa J Statist, (In press) 2 Rainville, ED (1960) Special Functions Chelsea Publication Co Bornmx US` 3 Roohi, (2003) Negative and Factorial Moments of Discrete Distributions Involving Hyper-Geometric Series Functions Unpublished PhD dissertation, National College of Business dministration and Economics, Lahore

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