Lowercase $x$ vs Uppercase $X$ in statistics

Question

There are many definitions online but they do not seem to align, or at least do not clearly correspond to, the usages I see elsewhere. For example: this website defines the difference is being $X$ refers to a set of population elements; and $x$, to a set of sample elements. $N$ refers to population size; and $n$ to sample size.

However in this post we see:

Now consider $X \sim U(0,1)$ and the sequence of random variables $X_n = \left(1 + \frac{1}{n}\right) X$. This is a sequence of RVs with $X_1 = 2X$, $X_2 = \frac{3}{2} X$, $X_3 = \frac{4}{3} X$ and so on. In what senses can we say this is getting closer to $X$ itself?

I do not understand what $X$ and $n$ mean here. I thought $X$ was a function that mapped from the sample space $\Omega$ to the reals? If that's the case what does $X_n$ mean here? Are there $n$ (the size of the sample space) possible functions that map from $\Omega$ to the reals?

Answer 1

You are correct that $X$ is a Borel function $X : \Omega\to\mathbb R$. A realization of $X$ is a particular real number $X(\omega)$, and such a number may be referred to as $x$. This means, for example, the event $\{X = x\}$ is more formally written as $\{\omega\in\Omega : X(\omega) = x\}$.

We can define as many functions as we want from $\Omega$ to $\mathbb R$, so e.g. we can imagine having $X_1, X_2, \dots : \Omega\to\mathbb R$. If $X_i = a_i X$ for constants $a_i\in\mathbb R$ then each function $X_i$ is also Borel so these are all random variables too. $n$ is often used to index a sequence of random variables so I could refer to $X_n$ as an arbitrary element of the sequence of functions $X_1, X_2, \dots$ instead of using $X_i$ or some other indexing variable. This use of $n$ is totally different from that of a sample size, although often the sequence of RVs in question comes from imagining taking bigger and bigger samples so $X_n$ may correspond to a sample of $n$ things, but it doesn't have to.

We can then ask about how the sequence of RVs evolves by letting $n \to \infty$. For example, in your case since $a_n \to 1$ as $n\to\infty$ we'll have $X_n$ converging on $X$ in some sense.

Answer 2

In this context, capitalisation is usually used to distinguish random variables from fixed values, so you are correct to interpret $X: \Omega \rightarrow \mathbb{R}$ as a random variable. The post specifies that it is considering a sequence of random variables $X_n : \Omega \rightarrow \mathbb{R}$, so there are an infinite number of these random variables, of the form:

$$\begin{align} X_1(\omega) &= (1+\tfrac{1}{1}) X(\omega), \\[6pt] X_2(\omega) &= (1+\tfrac{1}{2}) X(\omega), \\[6pt] X_3(\omega) &= (1+\tfrac{1}{3}) X(\omega), \\[6pt] &\ \ \vdots \\[6pt] X_n(\omega) &= (1+\tfrac{1}{n}) X(\omega), \\[6pt] &\ \ \vdots \\[6pt] \end{align}$$

Finally, the value $n$ here has nothing to do with the size of $\Omega$ or any other substantive property of the sample space. It is merely an index used to specify the form of the random variables in the sequence --- the post could just as easily have used the index $i$, or any other variable notation here.