If X~NB(r,p) then find the PDF of Y = X - r.
The answer is:
[ (y+r-1) , (r-1) ] p^r(q)^y ; y=0,1,2,...
I have no idea how to get to that. Any help would be appreciated.
The answer is:
[ (y+r-1) , (r-1) ] p^r(q)^y ; y=0,1,2,...
I have no idea how to get to that. Any help would be appreciated.
-
x=w(y) = y+r
dw(y) / dy = 1
The PDF for a negative binomial is [ (x-1) , (r-1) ] p^r q^x-r
Thus, the formula for the pdf of y is fx(y+r) |dw(y)/dy|
So the second part is 1, so that just disappears. Finally, in the first part, all you need to is plug in y +r wherever you see x.
Thus, you get: [ (y+r-1) , (r-1) ] p^r(q)^y
Hope that helps!
dw(y) / dy = 1
The PDF for a negative binomial is [ (x-1) , (r-1) ] p^r q^x-r
Thus, the formula for the pdf of y is fx(y+r) |dw(y)/dy|
So the second part is 1, so that just disappears. Finally, in the first part, all you need to is plug in y +r wherever you see x.
Thus, you get: [ (y+r-1) , (r-1) ] p^r(q)^y
Hope that helps!