Probability density function

The Probability Density Function (PDF) gives information about how likely a value of a random variable is. It is not directly the probability but the probability per unit length. For a given random variable X we define the PDF $p_{x} (x)$ as follows:

p_{X} (x) = \lim_{∆ \to 0} \frac{P (x < X < x + ∆)}{∆} = \lim_{∆ \to 0} = \frac{F_{X} (x + ∆) - F_{X} (x)}{∆}

p_{X} (x) = \frac{d F_{X} (x)}{d x}

For step discontinuities in the CDF, the Dirac delta function is used to describe the PDF.

Properties:

$P (x \leq x_{0}) = F_{X} (x_{0}) = \int_{- \infty}^{x_{0}} p_{X} (x) d x$
$p_{X} (x) \geq 0, \forall x$
$\int_{- \infty}^{+ \infty} p_{X} (x) d x = 1$
$P (a < x < b) = \int_{a}^{b} p_{x} (x) d x$

The PDF of discrete random variables accumulate the probability in a finite and numerable number of values, which are the possible values. Its PDF can be expressed using the Dirac delta function:

p_{X} (x) = \sum_{i - 1}^{N} P (X = x_{i}) δ (x - x_{i})

Review of Basic Concepts of Probability

Probability density function

Properties: