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)(sumlimits_{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(egin{array}{c}{k+r-1} \ {k}end{array}
ight)=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(egin{array}{c}{-r} \ {k}end{array}
ight))
continuous distribution：
Normal distibution：(int_limits{mathbb{R}} exp left(-frac{x^{2}}{2}
ight) mathrm{d} x=1)
(int_{0}^{infty}exp left(-frac{x^{2}}{2}
ight) mathrm{d} x=frac{1}{2})
(X looparrowright N(mu,sigma^2))
pdf：(p(x)=frac{1}{sqrt{2pi}sigma}e^{frac{-(x-mu)^2}{2sigma^2}})
cdf：(F(x)=frac{1}{sqrt{2pi}sigma}int_{-infty}^xe^{frac{-(t-mu)^2}{2sigma^2}}dt)