Probability Density

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.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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,…, x_s) for several random variables X₁, X₂,…, X_s is defined in a similar manner and is called the joint probability density. Thus, for any a_i and b_i, the probability that the inequalities a₁ < X₁ < b₁,…, a_s < X_s < b_s are simultaneously satisfied is equal to

If the random variables X₁, X₂,…, X_s 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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.