## Center of mass of normal random vector

All,

I'm facing with a simple probability problem.

I have a even symmetry function p[k],

p[k] = 0 if |k| > K. p[k] can be any finite even symmetry function.

And I have the same length of random noise n[k].

n[k] follows normal distribution with a variance $\sigma^2$.

Now I want to find a distribution of following;

$
t = \frac{\sum k \cdot d[k]}{\sum d[k]}\\
$

where $d[k]=p[k]+n[k]$. After MATLAB simulation, I realize that $t$ follows a normal distribution. But I cannot derive it.
Would you please give me a hint to solve this problem?