Discrete Conditional Distribution

Feb 2015
172
39
Upstate NY
Let X and Y be random variables with joint probability mass function pXY(x,y) =c(1-2^(-x))^y where X=1,2,3,...,N-1 and Y =1,2,3...

I need to find c.

All I can think of is the double summation Sum(y= 0 to oo) Sum(0 to N-1) (c(1-2^(-x))^y) but this does not seem doable.

I have no ideas what I can try next. Can I please get a hint.

Thanks!!!
 

romsek

MHF Helper
Nov 2013
6,665
3,002
California
$f_{XY}(x,y)=(1-2^{-x})^y,~~~x=1,2,\dots,N-1,~~y=0,1,2,\dots$

$\displaystyle{\sum_{x=1}^{N-1}~\sum_{y=1}^\infty}~f_{XY}(x,y) = \displaystyle{\sum_{x=1}^{N-1}~\sum_{y=1}^\infty}~(1-2^{-x})^y$

$\displaystyle{\sum_{x=1}^{N-1}}~2^x - 1 = 2^n-n-1$

so $c = \dfrac{1}{2^N - N - 1}$