2949775110

724

E[X^l(r, n, m, k)X^j(s, n, m, fc)] =

Cs-i

(r — l)!(s — r — l)!(m + 1)'

aV^+JC_s_ i

I    + 1), s — r — 1,7_S — 1)

i—1s-r-l

(3.9)

(r — l)!(s — r — l)!(m + l)^s" ( s — r — 1 \

:EEEE (-i)^p+,+“⁺”

p=0 g=0 u=0    t>=0

<J/f>)(_P) (»//%>

\    » j [o7»-«+p-0//3)][a7r-»+P + ?-{(i + j)/^}]’

/3 > max(i,j) and i, j = 0,1,2,____

Proof: From (1.6), we have

(3.10)

g[jy'(r,>i,m,i;)X^J(ii,rt,m,fc)] = _(r _ J_r- 1)1 J₀ J ^xVl^F(^X)T f(^x)

(r-l)!(

^x9m^_1(i^;’W)[ftm(F(!/)) - hm(iⁱ’(!i:))]'^-r-1[łⁱ(»)r'^_1/(w)<łl/<fa. On expanding g^^_1(i^r’(a:)) binomially in (3.11), we get

C,-!

(3.11)

.E[X^l(r, n, m, fc)X^J (s, n, m, fc)] = r- 1

(r — l)!(s — r — l)!(m + l)^r_ Ji,j{m + u(m + 1), s — r — l,7s — 1).

Making use of the Lemma 3.2, we derive the relation in (3.10). Identity 3.1: For 7r,7s > 1, k > 1, 1 < r < s < n and m^-1

E i-D”

1    _ (s — r — l)!(m + l)^s‘

7—«

Ilter+1 7>

(3.12)

rE E I-D”

Proof. At i = j = 0 in (3.10), we have C_s-i

/ r-1 \ / s — r — 1 \    1

V ^U ) V ^v ) ls-vlr-u '

Now on using (2.13), we get the result given in (3.12).

At r = 0, (3.12) reduces to (2.13).

Special cases:

i) Putting m = 0, k — l in (3.10), the explicit formula for the product moments of order statistics of the type II exponentiated log-logistic distribution can be obtained as

E(x;_!„xL) = aV«c_r,.,„ EEEff-i)^ ( ⁿ~^s )

[a(n - s + 1 + v) + p - O'//³)]

[a(n -r + \+u)+p + q-{(i + j)//3}]'

(i/Wli)_