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The expected value of a random variable; or a location measure for a sample.

The mean of a probability distribution is also called its expected value. For a discrete random variable, $X$, its defined as:

$$E[X] = \mu = \sum_{x}x P(X=x)$$

Where $P(X)$ is the probability mass function and the sum is taken over all values that $X$ can take. For a continuos RV simply replace the summation with an integral.

The mean of a sample of points $(x_1, ..., x_n)$, also known as the sample mean, is the arithmetic average of all points, defined as:

$$\bar{x} = \frac1n \sum_{i=1}^{n}x_i$$

In literature, we reserve the use of $\mu$ to denote the true population mean and $\bar{x}$ as the mean of a sample of points, from some population.

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