Cumulative distribution function

Definition

• For both Discrete random variable and Continuous random variable
• $F(x)=P(X\le x)$ for $-\infty <x<\infty$
  • $F$ is non-decreasing function
  • ${\lim }_{x\to \infty }F(x)=1$
  • ${\lim }_{x\to -\infty }F(x)=0$
  • $F$ is right continuous. That is, for any $b$ and any decreasing sequence $b_{n,}n\ge 1$, that is converges to $b$
    • ${\lim }_{x\to \infty }F(b_n)=b$