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Math for Quantitative Finance

CONTINUOUS RANDOM VARIABLES AND PDFS

A random variable X is called continuous if there is a non-negative function f_x , called the probability density function of X, or pdf such that $$ P (X \in B) = \int_B f_X(x) dx $$, for every subset B of the real line. In particular, the value of X falls within an interval is

$$ P (a \leq X \leq b) = \int_a^b f_X(x) dx $$

PDF fx follows following properties $$ f_X(x) \geq 0 \space for \space all \space x \space and $$

$$\int_{-\infty}^{\infty} f_X(x) dx = 1$$