I'm calculating the problem descripted in the title, and found it a little bit hard, here is the problem:
Suppose $X_i\sim\mathcal{N}(0,1)$ is standard normal random variables, now we need to calculate the expectation of linear combination of the square of standard normal variables, that is $$\mathbb{E}\left[\sum_{i=0}^{L}c_i X^2_i\right].$$
There is a paper [1] giving a more general circumstance but the result is too complicated to calculate expectation. Another paper [2] gives a subtle theorem to derive a very simple result but it is conditional on a complicated condition.
Have you any idea to solve it?
[1]:Moschopoulos, P. G.; Canada, W. B., The distribution function of a linear combination of chi-squares, Comput. Math. Appl. 10, 383-386 (1984). ZBL0576.62022.
[2]:Fleiss, J. L., On the distribution of a linear combination of independent chi squares, J. Am. Stat. Assoc. 66, 142-144 (1971). ZBL0218.62014.
$$\mathbb{E}\left[\sum_{i=0}^{L}c_i X^2_i\right] = \sum_{i=0}^{L}c_i\mathbb{E}\left[ X^2_i\right] = \sum_{i=0}^{L}c_i,$$ where the first equality comes from linearity of expectation and the second from the fact that your random variables are zero mean and variance 1.