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I just picked up a book, and while I was skimming through it one statement caught my eye. In Chapter 2, page 47 of Statistical Inference it reads:

If X is a random variable with cdf $F_X(x)$ then any function of X, say g(X), is also random variable.

I am not quick to assume that book is in the wrong, but it is the case now isn't it? It should only hold for discrete $X$ or by adding assumption that $g$ is measurable.

A bit off topic: this book is recommended a lot and presented as rigorous (for example here), but all the books I've read on stats completely disregard measure theory and skip technical aspects. Is it just a normal approach in statistics and I am not missing out by going with such books, or should i get something more technical?

Kombajn
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    Yes, many Statistics books disregard measure theoretic technicalities and that is sad. – geetha290krm Jan 24 '24 at 23:03
  • There are rigorous stats books. For example, the book by Jun Shao or the many by Le Cam / Lehmann – Andrew Jan 25 '24 at 00:39
  • It is certainly true if $g:\mathbb{R}\rightarrow\mathbb{R}$ is measurable. If $g$ is not measurable, your statement about the discrete case hinges on the definition of "discrete." It is true if you mean a random variable $X:\Omega\rightarrow\mathbb{R}$ is "discrete" if its image $X(\Omega)$ is a finite or countably infinite set. But then you cannot necessarily call a random variable $X:\Omega\rightarrow\mathbb{R}$ that is almost surely 0 "discrete." – Michael Jan 25 '24 at 16:06
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    I observe in your book link, the Definition 1.4.1 on page 27 says "A random variable is a function from a sample space S into the real numbers." So you may as well complain about that definition rather than its consequence on page 47. The author likely wanted to avoid distractions about sigma algebras and measurability, a lot of famous books do that. Most functions $g:\mathbb{R}\rightarrow\mathbb{R}$ of practical interest satisfy the "measurable" assumption, including all functions that have at most finite or countably many discontinuities, so you are unlikely to encounter nonmeasurable ones. – Michael Jan 25 '24 at 16:29
  • @Michael I noticed that too a while after made the post, after tracing back the definitions. I guess thats an oversight from my side, I got too used to measure theoretic definition of random variable and assumed that it will be the case at all times. – Kombajn Jan 25 '24 at 21:21

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Jack Cohen and Ian Stewart once discussed the concept of "lies-to-children" - an incorrect presentation of a technical concept that is "good enough" to provide an understanding of how something works while still being, fundamentally, a lie. The classic example is the Rutherford-Bohr model of an atom being like a tiny solar system as compared to our current understanding of it as a weird cloud of probability wavelets.

In mathematics, there are also plenty of lies-to-children. In arithmetic we are gradually introduced to negative numbers, fractions, irrationals and complex numbers, each time being told "you can't perform this operation" (like subtracting or dividing bigger numbers from smaller) before being shown a system where you can.

By making a blanket statement like "any function of a random variable is a random variable", and by not discussing measurability at all, Casella and Berger are presenting another lie-to-children, even if the children in this case have a university level of education.

Is it reasonable? That's a tough question. Given the level of complexity of the text, going into any depth of measurability, or a formal construction of random variables, would probably be inappropriate. Covering the topic properly would be a text unto itself (and of course to some extent it's "turtles all the way down" because a formal dive into measure theory intersects with calculus, and needs a certain amount of set theory, at which point we probably need to dip into the ZFC axioms and decidability). But could they have mentioned it somehow?

It might have been nice to have a brief mention - just something that says "There is an even more rigorous treatment of random variables, but for the purposes of this text consider every theorem to have an implicit 'if everything exists' added to it".

If your aim is to understand core concepts in statistics and apply them in any kind of practical situation, this book is perfectly suitable. It covers a huge number of topics with proofs and working, and there are millions of people using these concepts in the real world who wouldn't know a Borel algebra if it slapped them on a set of finite measure. On the other hand, if you are still itching to get under the hood and understand what makes all of this work (and perhaps more interestingly the situations where it doesn't work) then you will need a new text.

While I can't offer an actual opinion on their quality, here are three texts that appear to cover the subject at the required level of rigor:

ConMan
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  • +1. I'll add Shao's Mathematical Statistics to the list of good rigorous treatments of statistical theory – Jose Avilez Jan 25 '24 at 03:46
  • Here is another soft "lie" that you likely will find even in measure theory books: If we define a random variable as a measurable function $X:\Omega\rightarrow\mathbb{R}$, then if we consider a probability experiment with infinite i.i.d. coin flips, if we define $X$ as the number of flips until we first get heads, many would say this is a $Geom(1/2)$ random variable: but of course we might have a (prob zero) outcome $\omega^$ for which $X(\omega^)=\infty$, so strictly speaking $X$ may need to be called an "extended" random variable. – Michael Jan 25 '24 at 16:23