The sum of standard deviations of a set of bounded variations

Question

Suppose we have $n$ nonnegative random variables $\{X_i\}_{i=1}^n$, and we have $X_1+X_2+\cdots+X_n=c$ is a constant.

How can we get an upper bound for $\sigma_1+ \sigma_2+\cdots+\sigma_n$, where $\sigma_i$ is the standard deviation of $X_i$? (Note the distribution is unknown.)

Thanks!

Answer 1

Hint: We can find from the sum that $\mathbb{E}(X_1)+\cdots+\mathbb{E}(X_n) = c$. If they have the same distribution we can find that $\mathbb{E}(X_i) = \frac{c}{n}$. Now, You can using from Chebyshev's inequality to find an upper bound with a specified probability.