 # integration – fractional moments of multivariate regular distributions Answer

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integration – fractional moments of multivariate regular distributions

Mathematica offers some outcomes for the bivariate regular distribution with correlation $$rho$$ and benchmark marginals. If $$a$$ and $$b$$ are constructive integers then

commence{align} textual content{for queer }a+b, E[x^a y^b] &=0

textual content{for queer }atext{ and }b, E[x^a y^b] &=
sqrt{frac{2^{a+1}}pi}
Gamma!left(1+frac a2right)
f(a,b,rho)

textual content{for plane }atext{ and }b, E[x^a y^b] &=
sqrt{frac{2^{a}}pi}
Gamma!left(1+frac a2right)
f(a,b,rho)
aim{align}
the place
commence{align} f(a,1,rho)&=rho f(a,3,rho)&=3rho+(a-1)rho^3 f(a,5,rho)&=15rho+10(a-1)rho^3+(a-1)(a-3)rho^5 f(a,2,rho)&=1+arho^2 f(a,4,rho)&=3+6arho^2+a(a-2)rho^4 f(a,6,rho)&=15+45arho^2+15a(a-2)rho^4 + a(a-2)(a-4)rho^6 aim{align}
For queer $$a$$ and $$b$$ the coefficients emerge to breathe Ward numbers, and for plane $$a$$ and $$b$$ they emerge to breathe the exponential Riordan array.

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