Before the realizations of the random variables $X_1,\dots,X_n$ become known, their sample mean can be regarded as a random variable.
This statement confuses me. Can you give me an example?
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2$\begingroup$ What is the average of three rolls of a die? It's a function of the individual rolls which are themselves random, and so the average is also random. $\endgroup$– dsaxtonSep 14, 2016 at 16:45
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1$\begingroup$ You don't want to pass on a beautifully detailed explanation on this topic by @fcop $\endgroup$– Antoni ParelladaSep 14, 2016 at 17:17
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$\begingroup$ @dsaxton oh, i see. so this means that even if we use realizations of the rvs, sample mean is itself a random variable, right? i'm not too familiar with mathematical notation, thats why im asking. $\endgroup$– John RichardSep 14, 2016 at 17:26
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$\begingroup$ Once you roll the die then you know what value it takes on and it's no longer random. But when we talk about the average beforehand, it is just as random as the individual rolls. $\endgroup$– dsaxtonSep 15, 2016 at 2:44
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$\begingroup$ @JohnRichard John, if you feel like I answered your question, please consider "accepting" my answer by clicking on the tick mark on the left. $\endgroup$– Antoni ParelladaSep 17, 2016 at 6:10
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1 Answer
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Without mathematical notation, and to elaborate on @dsaxton didactic comment...
In a probability "problem" (read, space) there is a random experiment. This is something that happens out there in the physical world: the tossing of a coin, the rolling of a die, or the temperature in Philadelphia in the morning. For God (sorry, just a construct to move the argument along) the experiment is not random; and it wouldn't be random for us if we could account for every single minute factor that comes into play (direction of the wind, force exerted on the die, exact material composite of the die, height reached upon tossing, acceleration,... you get the gist). But all we can do is observe.
What we observe is the outcome. And we assign (read, "map") this outcome to a numeric value in the real line. This function from the observation of something happening to the assignment of a numerical value to it is the random variable. And the map or function would be defined like this: "we look at the die, and we notice which side is facing up." Yes, kind of silly, but we could have decided to note the velocity of the die upon impacting the table - that would be a different random variable (a continuous one) on the same random experiment: rolling the die.
Now we have a the random variable defined. The actual realizations, say $1$ or $6$, are not the random variable - they are individual outcomes.
A different random variable that we can choose to define is on a sample - not a single experiment, but a group of experiments: multiple die rolls, say. We can define it as: "Note the number facing up in $5$ die rolls; add them; and divide by $5$". The sample mean.
This is another random variable - a statistic on the sample.
They are related, and when we have a realization of the group of experiments ($5$ die rolls), say, $\{5, 5, 6, 1,2\}$ the mean will be determined - no longer a random variable. I believe that is what (loose terminology aside) the sentence: "Before the realizations of the random variables $X_1,\dots,X_n$ become known, their sample mean can be regarded as a random variable" implies.
The sample mean is just a statistic tying this answer to the more rigorous post by @fcop. To quote a key passage of his/her answer:
So, to answer your question, a test statistic is a special case of a random variable. The test statistic is thus a random variable, related to the random variable from which the sample (used to compute the test statistic) was drawn.
answered Sep 14, 2016 at 17:45
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$\begingroup$ This makes sense. Thank you so much, thank you everyone for their answers. $\endgroup$ Sep 14, 2016 at 18:17
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$\begingroup$ @John Richard I'm happy it made sense - it is a difficult and fascinating topic as you can see on the post I link. I did my best to explain it stripped of baffling set theory notation. $\endgroup$ Sep 14, 2016 at 18:19