Probability density function

In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. A probability density function is everywhere non-negative and its integral from −∞ to +∞ is equal to 1. If a probability distribution has density f(x), then intuitively the infinitesimal interval has probability f(x) dx. Informally, a probability density function can be seen as a "smoothed out" version of a histogram: if one empirically measures values of a continuous random variable repeatedly and produces a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density (assuming that the variable is sampled sufficiently often and the output ranges are sufficiently narrow).

Related Topics:
Mathematics - Probability distribution - Integral - Interval - Histogram - Continuous - Random variable

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Formally, a probability distribution has density f(x) if f(x) is a non-negative Lebesgue-integrable function R → R such that the probability of the interval is given by

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:int_a^b f(x),dx

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for any two numbers a and b. This implies that the total integral of f must be 1. Conversely, any non-negative Lebesgue-integrable function with total integral 1 is the probability density of a suitably defined probability distribution.

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