The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by X{\displaystyle X}, then it is also known as the expected value of X{\displaystyle X} (denoted E(X){\displaystyle E(X)}). For a discrete probability distribution, the mean is given by ∑xP(x){\displaystyle \textstyle \sum xP(x)}, where the sum is taken over all possible values of the random variable and P(x){\displaystyle P(x)} is the probability mass function. For a continuous distribution, the mean is ∫−∞∞xf(x)dx{\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx}, where f(x){\displaystyle f(x)} is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite (+∞ or −∞), while for others the mean is undefined.

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