nLab empirical mean

Redirected from "empirical means".
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Idea

In probability theory, the empirical mean or empirical average is the average value of a quantity, with weights given by empirical frequencies.

For example, if we roll a die 3 times, and we obtain first 22, then 33, then 55, the empirical mean is

2+3+53=1033.33. \frac{2+3+5}{3} \;=\; \frac{10}{3} \;\approx\; 3.33 .

The name empirical mean denotes both the distribution obtained by sampling a finite amount of data, as well as the limit (when it exists) resulting from an infinite sequence of observations, usually generated from a stochastic process.

In statistics it is used as an estimator? of the expectation value of a random variable whenever it is possible to take iid samples.

In measure-theoretic probability

Let NN be a finite set. We can view the product space N\mathbb{R}^N as the space of finite sequences (x 1,,x n)(x_1,\dots,x_n) of real numbers. The empirical mean of a finite sequence (x 1,,x n) N(x_1,\dots,x_n)\in \mathbb{R}^N is the average

x 1++x nn. \frac{x_1+\dots+x_n}{n} .

Similarly, we can view the countable product \mathbb{R}^\mathbb{N} as the space of infinite sequences (x 1,x 2,x 3)(x_1,x_2,x_3\dots) of real numbers. The empirical mean of a sequence (x 1,x 2,x 3,) (x_1,x_2,x_3,\dots)\in \mathbb{R}^\mathbb{N} is the limit, if it exists,

lim n1n i=1 nx i. \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n x_i .

If the x ix_i are random variables, and so they form a stochastic process (for example, if they are coin flips), the empirical distribution, if it exists, is a random variable as well.

In categorical probability

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Properties

See also

References

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category: probability

Created on July 15, 2024 at 16:44:05. See the history of this page for a list of all contributions to it.