Fungsi dénsitas probabilitas

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Dina matematika, probability density function dipake keur ngagambarkeun probability distribution di watesan integrals. Lamun probability distribution ngabogaan densiti f(x), saterusna interval tak terhingga [x, x + dx] ngabogaan probabiliti f(x) dx. Probability density function bisa oge ditempo tina versi "smoothed out" histogram: if one empirically measures values of a variabel acak 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).

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 [a, b] is given by

$\int_a^b f(x)\,dx$

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.

Contona, sebaran seragam dina interval [0,1] ngabogaan probabiliti densiti f(x) = 1 keur 0 ≤ x ≤ 1 jeung nol dimamana. Standar sebaran normal ngabogaan probabiliti densiti

$f(x)={e^{-{x^2/2}}\over \sqrt{2\pi}}$ .

Lamun variabel acak X diberekeun sarta distribusina kaasup kana fungsi probabiliti densiti f(x), mangka nilai ekspektasi X (lamun eta aya) bisa diitung ku

$\operatorname{E}(X)=\int_{-\infty}^\infty x\,f(x)\,dx$

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

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 one-point 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 - consider a random variable with a uniform distribution between 0 and 1/2.

Dua densiti f jeung g for the same distribution can only differ on a set of Lebesgue measure zero.

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