I have a set of random variables $x_k$, $k\in(1\cdots N)$, which are log-normally distributed (with standard parameters $\mu$ and $\sigma$). I would like to find expressions for $\sum_{k=1}^N \sqrt{x_k}$ and $\sum_{k=1}^N x_k$.
$N$ is large so, if I understand correctly, I can model the frequency density of $x$ with the function $f(x)=N\cdot \mathrm{pdf}(x; \mu, \sigma)$, where $\mathrm{pdf}(x; \mu, \sigma)$ is the probability density function for the log-normal distribution with parameters $\mu$ and $\sigma$. If that is correct, I think I can calculate the sum of $x_k$ as approximately $N\int_0^\infty x\cdot \mathrm{pdf}(x; \mu, \sigma)\ dx$, and the sum of $\sqrt{x_k}$ as approximately $N\int_0^\infty \sqrt{x}\cdot \mathrm{pdf}(x; \mu, \sigma)\ dx$. I believe these are just equal to the 1-st and 0.5-th arithmetic moments of the log-normal distribution, as given in this wikipedia article.
Is this logic correct?