Probability density function

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A probability density function, P(x), is a function representing the probability of a continuous random variable, x, (occurring). More accurately

\mbox{Prob}(a<x<b<)=\int_a^b P(x)\;dx.

If x is a continuous random variable then, of course, the probability that x=a is zero for any a. This is seen in the above as b \to a.

A random variable that is discrete may be included in this framework via the use of the Dirac delta function to represent a finite probability.

For obvious reasons P(x) must satisfy the following properties:

  • P(x)\ge 0
  • \int_{-\infty}^\infty P(x)\;dx=1.
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