## Finding the Fisher's information of this distribution

Suppose X1, X2...Xn are independent, each with density

$\displaystyle f(x;a) = 1/(\pi(1+(x-a)^2)$

where x, a are in R.

Essentially I've got to the stage in a question where I need to show that Fisher's information for a is n/2.

I seem to be getting that it should be 2n, so am wondering if I am making a mistake or if there is an error in the question.

I've got that the log likelihood is

$\displaystyle -n\log\pi - \sum \log(1+(x_i-a)^2)$

So differentiating twice and introducing a minus sign I've found the observed information (after simplifying):

$\displaystyle \sum(2-2(x_i-a)^2/(1+(x_i-a)^2)^2)$

To find the expected value of this (Fisher's information) I have multiplied by the pdf and integrated, as normal, to get that the Fisher's information for one observation is

$\displaystyle 4/\pi \int 1/(1+s^2)^2 ds$

where the integral is over -infinity to infinity. This gives an answer of 2 as the integral evaluates to pi/2.

So the Fisher's information of the sum is 2n, not n/2.

Have I made a mistake somewhere above? Thanks.