1 Attachment(s)

[SOLVED] Integration of Incomplete Beta Functions and Hypergeometric Functions

QUESTION

Let

I = double integral {x<y} [F(x)]^r * [F(y)]^s * [F(x)] * [1 - F(y)] dx dy

Verify/Show that for the 5 parameter Generalized Lambda Distribution (GLD),

I =

(a^2)(b^2)

--------------------------------------------

(r + b + 1)(r + s + 1 + 2b)(r + s + 2 + 2b)

abcd Gamma(d) Gamma(2 + r)

+ --------------------------------------------

(s + b) (s + 1 + b) Gamma(2 + r + d)

abcd Gamma(d) Gamma(r + s + 2 + b)

- --------------------------------------------

(s + b) Gamma(r + s + 2 + b + d)

abcd Gamma(d) Gamma(r + s + 3 + b)

+ --------------------------------------------

(s + b + 1) Gamma(r + s + 3 + b + d)

abcd Gamma(d + 1) Gamma(r + s + 2 + b)

+ --------------------------------------------

(r + b + 1) Gamma(r + s + 2 + b + d)

(c^2)(d^2) Gamma(1 + 2d) Gamma(r + 2) * F(-2,1+d,1+2d;2+d,r+3;1)

+ -------------------------------------------- -------------------------

(1 + d) Gamma(3 + r + 2d)

where b > -1/2 and d > -1/2

************************************************** *******

In the attachment I have provided the question with the all my working.

I am stuck at the integrations of the 3rd and 4th terms of my solution as I don't know how to integrate the incomplete beta function. Furthermore, their respective integrals I suspect should give 2 terms for a total of 4 for 'I' if you include the ones I have worked out. Yet in the question, 'I' is made up of 6 not 4 terms.

Also how does the hypergeometric function factor in the integration?

Finally is it possible there is an error in this question? I made sure that is was copied correctly.

Any hints, suitable resources or and help is greatly appreciated.

Thank you in advance