Is there a such thing as "standard error of random variable?"

Question

I am wondering if there is a such thing as "Standard Error" of a random variable? If so, is it simply just the standard deviation of the random variable?

Answer 1

As Benjamin Wang commented, the term "standard error" is usually reserved for random samples whose distribution is possibly unknown. Given $n$ independent sampled observations $o_1, \ldots, o_n$ with sample standard deviation $\sigma_o$, the sampled mean value $\bar{o}$ has a so-called "standard error on the mean" given by $$ \sigma_{\bar{o}} = \frac{\sigma_{o}}{\sqrt{n}} $$ where $\sigma_o$ is the sample standard deviation. See wiki for further details.

For random variables, the appropriate term you are probably looking for is just the standard deviation. This always exists for discrete random variables but may be undefined/infinite for continuous random variables. Classical examples where the standard deviation is undefined are the Cauchy distribution and the Pareto distribution for $\alpha \in (1,2]$.