Identical Random Variables

Question

I am reading the book "Probability - for the enthusiastic beginner" by David Morin.

The book makes the following statement about Identical random variables X_i.

" The sum X₁ + X₂ + X₃ + ..... X_n is not the same as nX. Although the random variables X_i are all identically distributed, that certainly doesn't mean that their values are identical. The values of the X_i will generally be different. "

As far as I know, Identical variables have the same values and probabilities so that they adhere to the same probability distribution.

So, I am finding it hard to accept what the book says about Identical random variables having different values.

Maybe my understanding of Identical random variables is wrong.

Can someone please explain to me with an example why Identical Random variables need not have same values ?

Answer 1

Identically distributed random variables have the same candidate values but not the same realizations, generally speaking. In the example below, $X$ and $Y$ are identically distributed: $\sim N(0,1)$.

X <- rnorm(1)

Y <- rnorm(1)

list(X = X, Y = Y)

$X [1] 1.234763

$Y [1] 1.186867

Answer 2

The key here is the difference between "identical variables" and "identically distributed variables". For a non-statistical example, imagine a factory making a particular model of car. Even if the manufacturing and assembly process remains identical (the same distribution), there will still be variability in the cars themselves (the values). If the process is tightly controlled, this will produce a distribution with little spread; if the process is less controlled, the distribution will be more spread out. Either way, some of the cars will be a number of millimeters (or microns if the process is even more tightly controlled) wider/narrower, longer/shorter, and higher/lower and a number of grams (or milligrams) heavier/lighter than typical (the mean could be used to describe the typical value for each of these attributes). You could take other measurements for colour, fuel economy, acceleration, etc. and the same would also apply.

Compare this with a science-fiction duplicator that produces atomic-level identical cars (values). They would simply be the same car (variable).

If we get ten cars from the factory, they will have a sum of their lengths (which won't be ten identical values providing we are using sufficiently precise measurements, setting aside the idea of quanta here limiting precision so that identical values could occur), a sum of their heights (the same), etc. (You could also look at means rather than sums here). The sum of the lengths of the ten cars will not necessarily be equal to the sum of the lengths of the first car and its nine duplicates.

Another way of looking at this is that the measurements from the factory car will have non-zero standard deviations, the measurements from the duplicated cars will have no variability whatsoever.