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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.

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  • $\begingroup$ But the mean is also (at least in some cases) the expected value. $\endgroup$
    – mavavilj
    Commented Dec 6, 2015 at 10:53
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    $\begingroup$ The situation in your question is not sufficiently clear. For example, if you're talking about sample means they can certainly be random variables. If you're talking about the population mean for a given random variable, then that's a fixed number (while it could be regarded as a degenerate random variable, it would be unusual to do so with a population parameter). Please clarify the circumstances in your question. $\endgroup$
    – Glen_b
    Commented Dec 6, 2015 at 11:02
  • $\begingroup$ I'm asking for "mean" in general. Which would ask for an answer considering also means that cannot be r.v.s. $\endgroup$
    – mavavilj
    Commented Dec 6, 2015 at 11:06
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    $\begingroup$ This is called conditional probability: the mean of $X$ may be $Y$ conditional on $Y$ and $Y$ still be a random variable. $\endgroup$
    – Xi'an
    Commented Dec 6, 2015 at 15:29

2 Answers 2

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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.]

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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.

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    $\begingroup$ Sorry, your answer sounds a bit vague to me. Can you show by means of an example? $\endgroup$
    – utobi
    Commented Jan 29, 2023 at 20:59
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    $\begingroup$ The mean of a sample is not the expected value of the variable. Perhaps you have misinterpreted the question? $\endgroup$
    – whuber
    Commented Jan 29, 2023 at 21:11

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