# Mean of root-sum-square of

Suppose that I have several normally-distributed random variables xi, each with its own different variance. All x's are zero-mean and independent. If y is the root-sum-square of the xi's, how do I find the mean of y?

• As noted, the problem is very simple when the variances are equal (look up Nakagami Distribution). When they are not, the problem is fairly complicated. See, e.g. this post. – steveo'america Jan 30 '19 at 21:21
• It seems like there might be a link to the Nakagami distribution even when the variances aren't the same? From Prof. Wikipedia, " ...accordingly, Nakagami-m is viewed as a generalization of Chi-distribution, similar to a gamma distribution being considered as a generalization of Chi-squared distributions." I've done a couple of test cases, and the fit to a Nakagami seems pretty good--now I just need a good explanation of the fit parameters. – Brian Jan 30 '19 at 21:56
• @steveo'america --I guess I don't understand how the linked post applies. I'm forming the square root of a sum of gamma-distributed RV's, not just the sum of gamma-distributed RV's. Or am I missing something? – Brian Jan 31 '19 at 19:21
• @brian : the linked post is for a simpler problem, but still very complicated. You have the further complication of taking the expected value of the square root of a sum of gammas. – steveo'america Jan 31 '19 at 20:15

This will involve application of the Law of the Unconscious Statistician here? We know that $$X_{i} \overset{id}{\thicksim} N(0,\sigma_{i}^{2})$$ for $$i=1,2,3,...,n$$, so the joint probability density function of $$X_1, X_2,...X_n$$ is given by:
$$\begin{eqnarray*} f_{X_{1},X_{2},...,X_{n}}(x_{,1},x_{,2},...x_{,n}) & = & \prod_{i=1}^{n}f_{X_{i}}(x_{i})\\ & = & \left(\frac{1}{\sqrt{2\pi}}\right)^{n}\prod_{i=1}^{n}\frac{1}{\sigma_{i}}e^{-\frac{x_{i}^{2}}{2\sigma_{i}^{2}}} \end{eqnarray*}$$
And by the Law of the Unconscious Statistician, if we set $$Y=h(x_{1},x_{2},...,x_{n})=\sqrt{\sum_{i=1}^{n}x_{i}^{2}}$$, then:
$$\begin{eqnarray*} E(Y) & = & \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}h(x_{1},x_{2},...,x_{n})\left(\frac{1}{\sqrt{2\pi}}\right)^{n}\prod_{i=1}^{n}\frac{1}{\sigma_{i}}e^{-\frac{x_{i}^{2}}{2\sigma_{i}^{2}}}dx_{1}dx_{2}\cdots dx_{n}\\ & = & \left(\frac{1}{\sqrt{2\pi}}\right)^{n}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\sqrt{\sum_{i=1}^{n}x_{i}^{2}}\prod_{i=1}^{n}\frac{1}{\sigma_{i}}e^{-\frac{x_{i}^{2}}{2\sigma_{i}^{2}}}dx_{1}dx_{2}\cdots dx_{n} \end{eqnarray*}$$