Probability density function
A probability density function serves to represent a probability distribution in terms of integrals. If a probability distribution has density f(x), then intuitively the infinitesimal interval [x, x + dx] has probability f(x) dx. A probability density function can be seen as a "smoothed out" version of a histogram: if one empirically measures values of a 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).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
- <math>\int_a^b f(x)\,dx</math>
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
- <math>f(x)={e^{-{x^2\over 2}}\over \sqrt{2\pi}}</math>.
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
- <math>E(X)=\int x\,f(x)dx</math>
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 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.)
Two densities f and g for the same distribution can only differ on a set of Lebesgue measure zero.
See also:
Fiorsen."
Fiorsen answered with a twisted, miserable smile:
"Poor wretches do what they can. May I see.html">see her? Let me just see
without his knowledge, keeping from him--HIM--all her feelings,
refreshment?"
Fiorsen shook his head, and there followed half an hour of acute
supported it better than his visitor. That child of nature, after
efforts with an expressive gesture, fidgeted here, fidgeted there,
stared out into the dark; came back as if resolved again to
the fire, flung himself down in an armchair, and turned his face to
writhings of this fellow who was endangering Gyp's happiness.
if not, why had she not told him? And he, too, suffered.
Then she came. He had expected her to be pale and nervous; but.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|