properties of continuous distributions

Let $S$ be a sample space. A continuous random variable is a map $X: S rightarrow R$ together with a function $f_X： R rightarrow R$ , having the properties that:

$$begin{aligned} (i) & f _ { X } ( x ) geq 0 text{ for all x real}\ (ii) & int_{-infty}^{infty} f_{X}(x)dx = 1\ (iii) & P[a leq X leq b]=int_{a}^{b} f_{X}(x) d x end{aligned}$$

The function $f_X$ is called the probability density function of the random variable $X$.

Cumulative Distribution:

$$F_X(x) = P[X leq x] = int_{-infty}^{x} f_{X}(t) d t$$

Obtain the density $f_X$ from $F_X$:

$$f_X(x) = F'_X(x)$$

$$mathrm{E}[X] =int_{-infty}^{infty} x cdot f_{X}(x) d x$$ $$E[varphi circ X]=int_{-infty}^{infty} varphi(x) cdot f_{X}(x) d x$$ $$operatorname{Var}[X] =mathrm{E}left[(X-mathrm{E}[X])^{2}right]=mathrm{E}left[X^{2}right]-mathrm{E}[X]^{2}$$