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I'm confused by some terminology. According to Casella and Berger (2002),

A random variable is a function from a sample space $S$ into the real numbers.

Let $X_1,...,X_n$ be iid $N(\mu, \sigma^2)$. In many texts, I see "random variable generation" as if values $X_1,...,X_n$ are randomly generated. But $X_i$ is not a value and can't be generated, can it?

I would think the term "random variate generation" is more reasonable in that $x_1, ... x_n$ is randomly generated.

What, then, does "random variable generation" mean? Is it different from "random variate generation"? Am I misunderstanding something?

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Random variate generation and random variable generation are used interchangeably. But as you can see from a google search, random variate generation is more common.

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