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 $2$, then $3$, then $5$, the empirical mean is
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.
Let $N$ be a finite set. We can view the product space $\mathbb{R}^N$ as the space of finite sequences $(x_1,\dots,x_n)$ of real numbers. The empirical mean of a finite sequence $(x_1,\dots,x_n)\in \mathbb{R}^N$ is the average
Similarly, we can view the countable product $\mathbb{R}^\mathbb{N}$ as the space of infinite sequences $(x_1,x_2,x_3\dots)$ of real numbers. The empirical mean of a sequence $(x_1,x_2,x_3,\dots)\in \mathbb{R}^\mathbb{N}$ is the limit, if it exists,
If the $x_i$ 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.
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The law of large numbers says that, under some conditions, the limiting empirical mean of an iid process equals the expectation value of the process.
More generally, the ergodic theorem? says that, under some conditions, the limiting empirical mean a random variable on an ergodic process equals its conditional expectation with respect to the invariant sigma-algebra.
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Created on July 15, 2024 at 16:44:05. See the history of this page for a list of all contributions to it.