统计学-数据分布

$$P(X=x|N) = frac{1}{N}$$
$$E(X) = frac{N+1}{2}$$
$$Var(X) = frac{(N+1)(N-1)}{12}$$

$$P(X=x|N,M,K) = frac{binom{M}{x}binom{N-M}{K-x}}{binom{N}{K}}$$
$$E(X) = frac{KM}{N}$$
$$Var(X) = frac{KM}{N}(frac{(N-M)(N-K)}{N(N-1)})$$

$$P(X=x)=p^x(1-p)^{1-x}$$
$$E(X)=p$$
$$ Var(X)=p(1-p)$$

$$P(X=x|n,p)=binom{n}{x}p^x(1-p)^{n-x}$$
$$E(X)=np$$
$$ Var(X)=np(1-p)$$

$$P(X=x|lambda) = frac{e^{-x}lambda^x}{x!}$$
$$E(X) = lambda$$
$$Var(X) = lambda$$

$$P(X=x|r,p)=binom{x-1}{r-1}p^r(1-p)^{x-r}$$
$$E(X)=frac{r(1-p)}{p}$$
$$Var(X)=frac{r(1-p)}{p^2}$$

$$P(X=x|p)=p(1-p)^{x-1}$$
$$E(X)=frac{1}{p}$$
$$Var(X)=frac{(1-p)}{p^2}$$

$$
f(x|a,b) =
begin{cases}
frac{1}{b-a}
0
end{cases}
$$
$$E(X)=frac{a+b}{2}$$
$$Var(X)=frac{(b-a)^2}{12}$$

$$Gamma(alpha)=int_{0}^infty t^{alpha-1}e^{-t}dt$$
$$f(x|alpha,beta)=frac{1}{Gamma(alpha)beta^alpha}x^{alpha-1}e^{-x/beta}$$
$$E(X)=alphabeta$$
$$Var(X)=alphabeta^2$$

• 当$alpha=p/2$且$beta=2$时，伽马分布为卡方分布
• 当$alpha=1$时，伽马分布为指数分布

$$f_Y(y|gamma,beta)=frac{gamma}{beta}y^{gamma-1}e^{-y^{gamma}/beta}$$

$$f(x|mu,sigma^2)=frac{1}{sqrt{2pi}sigma}e^{-(x-mu)^2/2sigma^2}$$
$$E(X)=mu$$
$$Var(X)=sigma^2$$

$$B(alpha,beta)=frac{Gamma(alpha)Gamma(beta)}{Gamma(alpha+beta)}$$
$$f(x|alpha,beta)=frac{1}{B(alpha,beta)}x^{alpha-1}(1-x)^{beta-1}$$
$$E(X)=frac{alpha}{alpha+beta}$$
$$Var(X)=frac{alphabeta}{(alpha+beta)^2(alpha+beta+1)}$$

$$f(x|theta)=frac{1}{pi}frac{1}{1+(x-theta)^2}$$

$$f(x|mu,sigma^2)=frac{1}{sqrt{2pi}sigma}frac{1}{x}e^{-(logx-mu)^2/2sigma^2}$$
$$E(X)=e^{mu+(sigma^2/2)}$$
$$Var(X)=e^{2(mu+sigma^2)}-e^{2mu+sigma^2}$$

$$f(x|mu,sigma)=frac{1}{2sigma}e^{-|x-mu|/sigma}$$
$$E(X)=mu$$
$$Var(X)=2sigma^2$$