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.
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