Probability Density Function

Probability Density Function#

For a continuous random variable \(X\) there may exist a probability density function (PDF) \(f_X\) which is the density of probability among its events. We can find the probability of a continuous event \(A = [a,b]\) by integrating over the PDF,

\[ P(a \le X \le b) = \int_a^b f_X ~\mathrm dx \]

Properties#

  • The axioms of probability applies:

    1. The PDF is non-negative,

      \[ f_X(x) \ge 0 \]

    2. The area of the PDF is norm-1.

      \[ \int_{-\infty}^\infty f_X(x)~\mathrm dx = 1 \]

Joint Density Functions#

The probability of two events,

\[ P(X \in A, Y \in B) = \int_B\int_A f_{XY}(x,y)~\mathrm dx \mathrm dy \]
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