Random variable vs Statistic? [duplicate]

What’s the difference between a random variable and a statistic?

It seems that formally, a random variable is simply any real-valued function (and its domain is a set that we call a “sample space”).

But isn’t the exact same also true of a statistic? That is, isn’t a statistic also simply any real-valued function (whose domain is a set called a “sample space”)?


Example. We roll a die. The sample space is $\Omega=\{1,2,3,4,5,6\}$.

Let $T$ be double the value of the die roll: Formally, $T:\Omega\rightarrow \mathbb R$ is the function defined by $T(x)=2x$.

Is $T$ a random variable, a statistic, or both?


In your answer, please give:

  1. A precise definition of a random variable
  2. A precise definition of a statistic
  3. An answer to the question in my example: “Is $T$ a random variable, a statistic, or both?”

Answer

A statistic is a function defined over one or more random variables.

So yes, a statistic is a random variable, and follows a distribution.

Another answer gave the example of the mean of a bunch of iid normal random variables.
$X_1,…,X_n\sim N(\mu,\sigma^2)$

The mean is a statistic because it is a function defined over random variables
$$\bar{X}= g(X_1, X_2 … X_n) = \frac{1}{n} \sum_{i=1}^n X_i $$

There is one condition however, which is that a statistic cannot explicitly depend on unknown parameters. Take the following definition of $g$ :

$$ g(X_1) = \frac{X_1 – \mu}{\sigma}$$

While $g$ here is a function of a random variable, and it follows a standard normal distribution, it’s not a statistic (unless $\mu$ and $\sigma$ are known).

For a more detailed explanation see pg. 122 of this.

Attribution
Source : Link , Question Author : JerryS1988 , Answer Author : Baba

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