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"Suppose we have a random sample of n independent and identically distributed random variables $X_i$" Hi, I am new to statistics. I am confused as to what $X_i$ represents. Are the $X_i$'s different variables? E.g. could $X_1$ be years of education, $X_2$ be age, etc?

Thanks :)

Glen_b
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Will
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1 Answers1

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No, because "Age" and "education" are not "identically distributed"

Consider instead a random sample of $n$ of those ages:

Age of first subject in sample = $X_1$
Age of second subject in sample = $X_2$
$\vdots$
Age of $n$th subject in sample = $X_n$

The material you quote is not well phrased and may be confusing if you don't already know what it's trying to say.

It seems as if it's referring to the entire collection as $X_i$, but in fact $X_i$ refers only to a particular one (the $i$th one). You can of course allow $i$ to take the values $1,2,...,n$ in turn, but that's not the same thing as being all of them at once (more typically you'd use bold or underline to indicate all of them together in a vector, say $\mathbf{X}=(X_1,X_2,...,X_n)^\prime$ ).

In a different situation to this, $X_1$ could represent a vector of ages, $X_2$ a vector of education-levels, and so on, and you'd refer to the individual values using $X_{1i}$, $X_{2i}$ and so on.

I have fudged/handwaved details of random variables a bit here but hopefully that gives a better sense of what that was talking about. You may like to read some of our questions and answers on random variables. whuber has a particularly simple and clear exposition here: What is meant by a random variable?

Glen_b
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  • This makes a lot of sense. But I thought that realisations of random variables were denoted by lower case letters? I thought X was the random variable and xi was the ith observation of X. – Will Aug 24 '19 at 12:18
  • I'm not referring to realizations at all in my answer. We're standing at the point before we actually see any observed values. Nonetheless we can talk about the first, the second, and so on. – Glen_b Aug 24 '19 at 12:22
  • Ahh, right. I really appreciate your time. Thanks! – Will Aug 24 '19 at 12:53
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    @Will usually $X_i$ means the ith variable, but some texts also use it to refer to the sequence of variables as a whole, depending on context. If they don't specify where the $i$ comes from (e.g. it's not the iterating variable of a $\sum$ or $\prod$) then the lack of specification is sort of taken to mean "all the possible things $X_i$ could actually be". Some other texts will be more explicit and write $\lbrace X_i \rbrace$ to refer to the sequence separate from any particular $X_i$. – Joseph Garvin Aug 26 '19 at 02:05