Mistake in Casella & Berger on page 207?

Question

Page 28:

A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable (or its range) will be denoted by the corresponding lowercase letters. Thus, the random variable X can take the value x.

Page 139:

For example, consider an experiment designed to gain information about some health characteristic of a population of people... the body weights of several people in the population might be measured. These different weights would be observations on different random variables, one for each person measured.

Page 207:

The random variables $X_1, ... , X_n$ are called a random sample of size n from the population f(x) if $X_1, ... , X_n$ are mutually independent random variables and the marginal pdf or pmf of each $X_i$ is the same function f(x).

Page 207 - I think the sentence below is incorrect:

Under the random sampling model each $X_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by f(x).

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First of all, given the authors' notation convention, we should rewrite this statement as follows, where the first $X_i$ has been replaced with $x_i$ because we are referring to a realized value of the random variable (an observation) and not the random variable itself.

Under the random sampling model each $x_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by f(x).

Secondly, as the authors themselves explained in the quoted text from page 139, $X_i$ is a different random variable from $X_j$ for all i$\neq$j. $X_1, ... , X_n$ each have the same distribution function (are identically distributed) but these are different random variables. For example, $X_1$ is the weight of the first person; $X_2$ is the weight of the second person, etc. It seems to me that in this sentence the authors have fallen into the trap of mistaking a random variable for its distribution. Or did I interpret things incorrectly? If this is a mistake, as I think it is, it is very unfortunate as it occurs in the paragraph explaining what a random sample is and this is precisely the place where the student is trying to get clarity on observations/random variables/distributions and how they interrelate in the case of a random sample.

Answer 1

That page describes identical and independently distributed variables $X_i$.

You could change that quoted piece in more ways. Probably the following would be better

Change

Under the random sampling model each $X_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by $f(x)$.

Into

Under the random sampling model each $X_i = x_i$ is an observation on the same population and each $X_i$ has a marginal distribution given by $f(x)$.

See also the piece on page 209

The random sampling model in definitions 5.1.1 is sometimes called sampling from an infinite population. Think of obtaining the values of $X_1, \dots, X_n$ sequentially. First, the experiment is performed and $X_1 = x_1$ is observed. Then, the experiment is repeated and $X_2 = x_2$ is observed. The assumption of independence in random sampling implies that the probability distribution for $X_2$ is unaffected by the fact that $X_1 = x_1$ was observed first. „Removing” $x_1$ from the infinite population does not affect the population, so $X_2 = x_2$ is still a random observation from the same population.

In that last sentence you see both the changes come together. We speak about the small letter when referring to the observation. So it is not $X_2$ that is called an observation but the observation that $X_2 = x_2$ is the observation. And it is not called 'an observation on the same variable' but 'an observation from the same population'.

Personally I think that it is not so bad to call the $X_i$ observations as well and the more important change in the quote is changing 'variable' into 'population'.

The $X_i$ can be seen as random variables that describe a random observation. Say, you could use it in a sentence as "we describe the random observations of the Donkey's walking speed with variables $X_1, X_2,\dots,X_n$". The small letter $x_i$ is more like the realisation of the observation rather than the observation itself. The small letter $x_i$ is not strictly the 'observation' itself but it is more like the 'observed value'. You see this also in the text with sentences like

$X_1 = x_1$ is observed

The $x_1$ is not the 'observation' itself (it is not the act) but it is the 'observed value', or it is 'what is observed'.

Answer 2

I think the source of confusion comes from the word observation and it's meaning in various contexts. I have to confess that this book was not my first statistical book read and perhaps this helped me to handle that easier.

The paragraph from Page 207 I understand it as:

Under the random sampling model each $X_i$ is an observable distinct random draw produced by the same phenomenon (statistical population described by an assumed variable) and each $X_i$ has a marginal distribution given by $f(x)$.

In different wording we can say that the random sampling model is the assumption that all random variables $X_i$ which are distinct have the same distribution because they are occurrences of the same phenomenon. Because they are produced by the same process they are observations of the same population. They still are distinct because they have independent error components.

This interpretation of word observation is consistent with surrounding paragraphs. For example:

The random sampling model describes a type of experimental situation in which the variable of interest has a probability distribution described by $f(x)$. If only one observation $X$ is made on this variable, then probabilities regarding $X$ can be calculated using $f(x)$.

and continues

In most experiments there are $n \gt 1$ ... repeated observations made on the variable, the first observation is $X_1$ ...

And immediately after your cited paragraph we have:

Furthermore the observations are taken in such a way that the value of one observation has no effect on or relationship with any other observations

The last paragraph is illuminating since it operates the distinction of the two components of an observation: it's value / outcome / data point and it's random variable. The paragraph states the value of one observation does not influence how other values are produced. It also states that the random variables which models each observation does not have any conditional leak of any sort.

In all those paragraphs other than the last one, they don't talk about values or outcomes. It talks only about how one can choose to model some random variables if we assume they are produced by the same phenomena. One cannot talk about any statistical property of a value. The value of an observation is dead from a probabilistic point of view. The associated random variable which produced that value can be used for that.

To me, a natural way to understand the word observation is as an unique occurrence, measurement or snapshot of some source which has an value where random noise is incorporated (observation result) which can be seen (if observable) or not (if hidden) and has an associated random variable (implicitly a probability distribution, sample space and so on) which describes the process. Sometimes I shortcut the terms and use observation to point to the random variables (like here in the random sample definition) and sometimes to point out the pieces of data (for example when we have a sample of that data). Somehow, it is still clear in my mind and well separated.

Later edit (I forgot to comment on your interpretation):

Under the random sampling model each $x_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by $f(x)$.

This looks wrong to me since if each $x_i$ is the outcome of the same variable ($X$ perhaps, certainly not any of $X_i$ since there are more than one variables there), than on which basis $X_i$ have the same marginal?