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 \leq X \leq b) = \int_{a}^{b} f_{X} d x

Properties#

The axioms of probability applies:
1. The PDF is non-negative,
  
  $f_{X} (x) \geq 0$
2. The area of the PDF is norm-1.
  
  $\int_{- \infty}^{\infty} f_{X} (x) d x = 1$

Joint Density Functions#

The probability of two events,

P (X \in A, Y \in B) = \int_{B} \int_{A} f_{X Y} (x, y) d x d y