Formally, a probability distribution has density f(x) if f(x) is a nonnegative Lebesgue integrable function R → R such that the probability of the interval [a, b] is given by
For example, the uniform distribution on the interval [0,1] has probability density f(x) = 1 for 0 ≤ x ≤ 1 and zero elsewhere. The standard normal distribution has probability density
If a random variable X is given and its distribution admits a probability density function f(x), then the expected value of X (if it exists) can be calculated as
Not every probability distribution has a density function; for instance the distributions of discrete random variables do not. A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous. In this case, F is almost everywhere differentiable, and its derivative can be used as probability density. If a probability distribution admits a density, then the probability of every onepoint set {a} is zero. (It is a common mistake to think of f(a) as the probability of {a}, but this is incorrect; in fact, f(a) will often be bigger than 1.)
Two densities f and g for the same distribution can only differ on a set of Lebesgue measure zero.
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