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Can the mean of some sample of random variables be itself a random variable?
Yes?
I think the particular confusion here is that since the calculation procedure for statistics is known, then they do not seem like "random draws". But they can take r.v.s as input and thus could be r.v.s. in the sense of being linear combinations of r.v.s, for example.
The mean of a collection of random variables is itself a random variable.
So if you have say, $X_1,X_2,X_3,...,X_n$, then $\bar{X}=\frac{1}{n}(X_1+X_2+X_3+...+X_n)$ is a random variable.
If you're talking about a sample of particular observations, $x_1, x_2, ..., x_n$, then you could look at the sample mean, $\bar{x}$ as another kind of observation, relating to the random variable $\bar{X}$ in the same way that the observation $x_1$ relates to the random variable $X_1$, which is to say it's a realization of the random variable $\bar{X}$.
[On the other hand, when we're talking about the mean of the distribution of a random variable, that is a parameter; it's not a variable $-$ at least not in a frequentist framework; it may be treated as one in a Bayesian framework, in which case it can be taken to represent the uncertainty in our belief/knowledge about its value.]
It's expected value of the random variable. Random variable itself is a function from sample space to real number. If some how one can define this function in a random way one may get random expected values. Given E(X) exist.
