I have a equation with a double sum. However, one of the variables in one of the sums comes from a stochastic distribution (Gaussian to be specific). How can I get a closed form equivalent of this expression? The $U$ and $T$ in the equation are constants.
$$ \sum_{k = 0}^{N_k-1} \bigg [ \big[ \sum_{i}^{N_i} \cos(\frac{4\pi}{\lambda} u_i k T) - \cos(\frac{4\pi}{\lambda} U k T) \big]^2 + \big[ \sum_{i = 1}^{N_i} \sin(\frac{4\pi}{\lambda} u_i k T) - \sin(\frac{4\pi}{\lambda} U k T) \big]^2 \bigg]$$
$$ u_i \thicksim \mathcal{N}(\mu_{u}, \sigma_{u}) $$
EDIT ========================================================
I have a simplified form now. It looks like the following
$$ \sum_{k = 0}^{N_k-1} \Bigg[ [N_i + 1] + 2 \sum_{i = 1}^{N_i} \bigg[ \sum_{m = 1}^{N_i} \cos(\frac{4\pi}{\lambda} (u_i - u_m) k T ) - \cos( \frac{4\pi}{\lambda} (u_i - U) k T) \bigg] \Bigg]$$
$$ u_i \thicksim \mathcal{N}(\mu_{u}, \sigma_{u}) $$ $$ u_m \thicksim \mathcal{N}(\mu_{u}, \sigma_{u}) $$
I want a function that looks like this $f(U, N_i, N_k, \mu_u, \sigma_u)$
-- following my comment above --
and then proceed with and then proceed with $$ \begin{array}{l} \sum\limits_{k = 0}^{N_k - 1} {\sum\limits_{i = 1}^{N_i } {\left( {2 - 2\left( {\cos \alpha \cos \beta + \sin \alpha \sin \beta } \right)} \right)} } = \\ = 2\sum\limits_{k = 0}^{N_k - 1} {\sum\limits_{i = 1}^{N_i } {\left( {1 - \cos \left( {\alpha - \beta } \right)} \right)} } = \\ = 2\left( {N_k N_i - \sum\limits_{k = 0}^{N_k - 1} {\sum\limits_{i = 1}^{N_i } {\cos \left( {\alpha - \beta } \right)} } } \right) \\ \end{array} $$
But the cosine of a (non-central) gaussian variable does not have a simple form (to my knowledge).
Unless you have that "most probably" $ u_i <<U$, in which case you can approximate the $\cos$ by series expansion.