2949775112

Solving the integral in I(x) by parts and substituting the resulting expression in (3.15), we get

E[X^l(r,n,m,k)X³ (s,n,m,k)\ — E[X^l{r,n,m,k)X³ (s — l,n,m,k)\

- -r.tr - ms¹- r -1)1 r r

x[h_m(F(y)) - h_m(F(x))]—'-¹[F(y)]'" d,ydx

the constant of integration vanishes sińce the integral in I(x) is a definite integral. On using the relation (1.3), we obtain

E[X^l(r, n, m, k)X³(s, n, m, k)] — E[X^l(r, n,m, k)X³ (s — 1 ,n,m, fc)] jaC_s.

a/57_s(r — l)!(s o/?7s(r — l)!(s — r — 1)!

«Wf(r))-L(W

and hence the result given in (3.14).

Remark 3.2 Setting m = 0, fc = 1 in (3.14), we obtain recurrence relations for product moments of order statistics of the type II exponentiated log-logistic distribution in the form

¹ - afln-. ₊!))*!*•”¹ =    + _af,<Ż-s₊l)^ElX^^]-

Remark 3.3 Putting m = — 1, k > 1 in (3.5), we get the recurrence relations for product moments of upper k records of the type II exponentiated log-logistic distribution in the form

Ratio moments of gos from type II exponentiated log-logistic distribution can be obtain by the following Theorem.

3.5. Theorem. For type II ezponentiated log-logistic distribution as giuen in (1.2)

E[X*(r,n,m,k)X^j ^(s,n,m,k)] =    -r—— -r-^-

UU pi9!r(i-p)r(i + i-_p)

(3.16)

(3.17)

^xn:,(i+^!±^)n;,_łl(i+“)' ^fi>3'

Proof From (1.6), we have

E[X^l(r,n,m,k)X^{3 p} (s,n,m,fc)] =

C_s-1

(r — l)!(s — r — l)!(m + l)^8-2

X f°° xⁱ[F(x)]⁽’-^r+*-°X^m+1)-¹f(x)J(x)dx, Jo