Probability Density Function
Contents
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:
The PDF is non-negative,
\[ f_X(x) \ge 0 \]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,
\[ P(X \in A, Y \in B) = \int_B\int_A f_{XY}(x,y)~\mathrm dx \mathrm dy \]