Probability Density


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probability density

[‚präb·ə′bil·əd·ē ‚den·səd·ē]
(quantum mechanics)
The square of the absolute value of the Schrödinger wave function for a particle at a given point; gives the probability per unit volume of finding the particle at that point.

Probability Density

 

The probability density of a random variable X is a function p(x) such that for any a and b the inequality a < X < b has probability equal to

For example, if X has a normal distribution,

If p(x) is continuous, the probability of the inequality x < X < x + dx is approximately equal to p(x) dx for sufficiently small dx. The probability density always satisfies the conditions

The probability density p(x1,…, xs) for several random variables X1, X2,…, Xs is defined in a similar manner and is called the joint probability density. Thus, for any ai and bi, the probability that the inequalities a1 < X1 < b1,…, as < Xs < bs are simultaneously satisfied is equal to

If the random variables X1, X2,…, Xs have joint probability density, they will be independent if, and only if, their joint probability density is the product of the probability densities of each of them.

References in periodicals archive ?
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As the probability density functions of considered distributions depend on the particular parameter values, construction of the classifiers should be started from the estimation of these parameters for each category of pixels.
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To solve related problems, we need to have a thorough understanding on the structure of the underlying probability density.

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