Distribution

Random Variable

\(\underline{cdf:}\)cumulative distribution function \(F(x)=P(X \leq x)\)
\(\underline{pmf:}\)probability mass function(for discrete probability distribution )
(1)\(p(x) \geq0,x \in X\)
(2)\(\sum\limits_{x \in X}P(x)=1\)
\(\underline{pdf:}\)probability density function(for continuous probability distribution )
(1)\(f(x) \geq 0\)for all x,
(2)\(\int_{-\infty}^{\infty}f(x)dx=1\)

discrete distribution:

Negative Binomial Distribution
\(\left(\begin{array}{c}{k+r-1} \\ {k}\end{array}\right)=\frac{(k+r-1) !}{k !(r-1) !}=\frac{(k+r-1)(k+r-2) \ldots(r)}{k !}=(-1)^{k} \frac{(-k-r+1)(-k-r+2) \ldots(-r)}{k !}=(-1)^{k}\left(\begin{array}{c}{-r} \\ {k}\end{array}\right)\)

continuous distribution：

Normal distibution：\(\int_\limits{\mathbb{R}} \exp \left(-\frac{x^{2}}{2}\right) \mathrm{d} x=1\)
\(\int_{0}^{\infty}\exp \left(-\frac{x^{2}}{2}\right) \mathrm{d} x=\frac{1}{2}\)
\(X \looparrowright N(\mu,\sigma^2)\)
pdf：\(p(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
cdf：\(F(x)=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^xe^{\frac{-(t-\mu)^2}{2\sigma^2}}dt\)