# The expectation of absolute value of the sum of n i.i.d. random variables

Let $$\varphi_i$$ be a Gaussian random variable such that $$\varphi_i \sim N(0,\sigma^2), \quad i = 1,2,\ldots,n.$$ What's the expectation: $$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |\right)$$ where $$\left | \cdot \right |$$ is the absolute value operation and $$j = \sqrt{-1}$$.

• I don't know if there is a nice closed form for the absolute value but for the square of the absolute value the expectation is $\frac{n (n-1)}{\exp \left(\sigma ^2\right)}+n$.
– JimB
Jan 1, 2019 at 19:55

The expectation you are interested in is equivalent to

$$E\left[\sqrt{n+2\sum_{j=1}^{n-1}\sum_{k=j+1}^n (\cos \varphi_j \cos \varphi_k+\sin\varphi_j \sin\varphi_k)}\right]$$

I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.

If you were interested in

$$E\left[n+2\sum_{j=1}^{n-1}\sum_{k=j+1}^n (\cos \varphi_j \cos \varphi_k+\sin\varphi_j \sin\varphi_k)\right]$$

then there is a closed-form solution:

$$n+n(n-1)e^{-\sigma^2}$$

• hi, since X is always positive, do we have $E[\sqrt{X}] = \sqrt{ E[X] }$? If it holds, the first expectation can have a closed-form solution, right?
– Vic
Nov 16, 2020 at 20:32
• @Vic No. If that were the case, then by your argument you'd have $E[X^2]=E[X]^2$ which would imply that all variances are identically zero.
– JimB
Nov 16, 2020 at 21:09
• that’s right. BTW, Jim, it seems that there is closed form solution for $$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |^{2}\right).$$ I am curious whether there is always closed-form solution or formula of expansion for $$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |^{2k}\right),$$ where $k$ is some integer.
– Vic
Nov 16, 2020 at 22:36
• @Vic You should ask that as a question.
– JimB
Nov 16, 2020 at 22:54
• Yes, that sounds a good idea. I have raised a question and considered a more general case. Jim, can you also have a look math.stackexchange.com/questions/3910426/…
– Vic
Nov 16, 2020 at 23:12