Find marginal pdf given joint pdf (stats)

[Phox]

Homework Statement

Given joint pdf:

f(x,y) = (2/(x²(x-1)))(y^{-(2x-1)/(x-1)}) x>1, y>1

find marginal pdf f_x(x)

Homework Equations

f_x(x) = ∫f(x,y)dy, 0, ∞

The Attempt at a Solution

f_x(x) = ∫(2/(x²(x-1)))(y^{-(2x-1)/(x-1)}) dy, 1, ∞

= [-2y^(x/(1-x))/x³],1,∞ = undefined. stuck here

 

[Ray Vickson]

Homework Statement

Given joint pdf:

f(x,y) = (2/(x²(x-1)))(y^{-(2x-1)/(x-1)}) x>1, y>1

find marginal pdf f_x(x)

Homework Equations

f_x(x) = ∫f(x,y)dy, 0, ∞

The Attempt at a Solution

f_x(x) = ∫(2/(x²(x-1)))(y^{-(2x-1)/(x-1)}) dy, 1, ∞

= [-2y^(x/(1-x))/x³],1,∞ = undefined. stuck here

No, it is NOT undefined. You have not used all the available information.

 

[Phox]

No, it is NOT undefined. You have not used all the available information.

have I bounded the integral correctly?

 

[Ray Vickson]

have I bounded the integral correctly?

I don't know what you mean. However, why make things complicated? When we do the y-integral, x is a constant, so we might as well write $f(x,y) = a y^{-b},$ where $a=2/[x^2(x-1)]$ and $b=(2x-1)/(x-1)$ are 'constants' as far as y-variations are concerned. You need to use information about the magnitude of $b$.