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I was looking at the following statement and felt I needed some help in sorting it out:

A random sample of size n from a population $f(x)$ is a collection of n independent random variables $X_1,...,X_n$, each having the distribution $f(x)$.

let us apply the above statement to all freshmen at a certain university where we are interested in their heights and weights, So I guess:

  1. $f(x)$ would be the population of freshmen at a certain university.
  2. What exactly does $x$ signify in $f(x)$
  3. so $X_1,...,X_n$ would be the weight measured for a sample of size $n$ ?
  4. so for heights would be need $Y_1,...,Y_n$?
  5. strictly speaking can we use $X$ for both height and weight (I guess no -- it would be meaningless as we're mixing types)?
  6. The last part of the statement "each having the distribution $f(x)$" has me a bit on tender ground -- would be interested in getting clarifications about that.

Note: The statement above is from: "Statistical Concepts and Methods (Wiley Series in Probability and Statistics), Gouri K. Bhattacharyya, Richard A. Johnson", page 208.

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  • $\begingroup$ The double use of $f(x)$ is confusing and should not have been used. Do you have a link? $\endgroup$ – Henry Oct 15 '12 at 7:21
  • $\begingroup$ This is from "Statistical Concepts and Methods (Wiley Series in Probability and Statistics), Gouri K. Bhattacharyya, Richard A. Johnson", page 208. $\endgroup$ – user1172468 Oct 15 '12 at 7:26
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The quotation you give is a definition. So it is saying that when the book says

a random sample of size $n$ from a population $f(x)$

this is shorthand (confusing in my opinion) for

a collection of n independent random variables $X_1,...,X_n$, each having the distribution $f(x)$

In this case $f(x)$ does not stand for the population, but for the distribution of the individual random variables. Since they are independent, either the random variables are continuous or the sampling is with replacement.

$x$ is a particular value the random variables can take. So for example you might write the probability one of them is less than or equal to $x$ as $P(X \le x)$.

Looking at your questions over weights, the population in this sense is not in fact the freshmen, but their weights, having the distribution $f(x)$. Another population in this sense is their heights, having a different distribution, and if you are talking about looking at both at the same time then your idea of using different letters is sensible.

But reading the surrounding paragraphs, I would comment that the whole section looks like an oversimplification likely to confuse. This may have come from trying to produce what the inside flap calls a "non–mathematical introductory statistics text", but it looks to me more like a lack of care in descriptions.

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  • $\begingroup$ I must say revisiting this topic many years after having studied it using a non mathematical introduction has been very confusing at best. $\endgroup$ – user1172468 Oct 15 '12 at 8:05

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