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The central limit theorem has an assumption like this "Let {X1, …, Xn} be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ^2." (quoted from wikipedia)

But in my understanding, a random variable should be something like "the Grade of a student" which can take on some values. "A sequence of random variables" is more like a random variable that takes on a sequence of values. e.g. Given a sequence of observations of Grade where Grade is one random variable.

So how do I make sense of this sequence of independent and identically distributed random variables.

Thank you so much for help!

1 Answers 1

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Consider, for example, an infinite sequence of coin-tosses with a fair coin.
$X_n = 0$ or $1$ depending on whether the $n$'th toss results in heads or tails.

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    Sorry I'm still confused: A random variable is defined by a function that associates with each outcome in Ω a value. Suppose we have {X1…Xn} random sample which contains iid random variables. Suppose we pick X1, I assume this is one observation. So how can one observation maps each outcome in the whole space to R? – 2017-02-19
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    The sample space for your sequence of random variables is typically not the same one that you might use for one random variable. An "outcome" in this case must give you the values of all the $X_i$. The sample space $\Omega$ might be the set $\{0,1\}^{\mathbb N}$ of all sequences of $0$'s and $1$'s. – 2017-02-19
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    I think I get your idea, thank you! – 2017-02-20