The content emphasizes the distinction between a random variable and its distribution. While a random variable measures a quantity depending on an outcome, its distribution reveals the long-term pattern of values over many repetitions. Changing the probability measure or the function defining the random variable can alter its distribution. Surprisingly, two random variables can have the same distribution but measure different things, illustrated by examples of tossing coins and births. The author highlights the importance of not confusing a random variable with its distribution, as seen in scenarios involving transformations and joint distributions, stressing the significance of understanding this difference to avoid common mistakes.
https://bookdown.org/kevin_davisross/probsim-book/distribution.html