Continuous random variable

Definition

• If there exists a nonnegative Probability density function $f$, defined for all real $x\in (-\infty ,\infty )$, having the property for any set $B$ of real numbers, $P(X\in B)={\int }_{B}f(x)dx$
• $P(a\le X\le b)={\int }_a^{b}f(x)$
• $P(X=A)={\int }_a^a=0$
  • Hence, $P(X\le a)=P(X\le A)$

Expected value:

• $E(X)={\int }_{-\infty }^{\infty }x.f(x)dx$
• $E(g(X))={\int }_{-\infty }^{\infty }g(x)f(x)dx$

Variance:

• If $X$ is a random variable with expected value, $\mu$
  • $Var(X)=E[(X-\mu {)}^2]=E[X^2]-(E[X]{)}^2={\int }_{-\infty }^{\infty }{x}^{2}f(x)dx-(E[X]{)}^{2}dx$
  • ${\sigma }_X=\sqrt{Var(X)}$
• Related variable:
  • $Var(aX\pm b)=a^{2}Var(X)$
  • $Var(X\pm Y)=Var(X)+Var(Y)$