离散数学

Markov’s inequality

设(X)为非负随机变量，则(foralllambda>0,Pr[Xgelambda]lefrac{mathrm E[x]}{lambda})。

Chebyshev’s inequality

设(X)为随机变量，则(foralllambda>0,Pr[|X-mathrm E[X]|gelambdasqrt{mathrm{Var}[x]}]lefrac1{lambda^2})。

Law of large numbers

设(X)为有限概率空间(Omega)上的随机变量，定义(Omega^n)上的随机变量(overline{X_n}=frac1nsumlimits_{i=1}X_i)，则(forallvarepsilon>0,limlimits_{n ightarrowinfty}Pr[|overline{X_n}-mathrm E[X]|>varepsilon]=0)。

Some Conclusions

Part 1

(forall tin[0,m],e^{-frac{t^2}{m-t+1}}lefrac{{2mchoose m-t}}{{2mchoose m}}le e^{-frac{t^2}{m+t}})
Proof:将(frac{{2mchoose m-t}}{{2mchoose m}})表达为(prod)形式，并运用(e^{1-frac1x}le xle e^{x-1})。

Part 2

(tgesqrt{mln C}+ln CRightarrowfrac{{2mchoose m}}{{2mchoose m-t}}ge C)
(tlesqrt{mln C}-ln CRightarrowfrac{{2mchoose m}}{{2mchoose m-t}}le C)

Part 3

(forall kin[0,m],sumlimits_{i=0}^{k-1}{2mchoose i}<2^{2m-1}frac{{2mchoose k}}{{2mchoose m}})
Proof:(frac{{2mchoose k-i}}{{2mchoose k}}lefrac{{2mchoose m-i}}{{2mchoose m}})

Part 4

(xsim B(2m,frac12),forall tin[0,m],Pr[x<m-tvee x>m+t]le e^{-frac{t^2}{m+t}})
(xsim B(n,frac12),forallvarepsilon>0,limlimits_{n ightarrowinfty}Pr[frac xnin[0.5-epsilon,0.5+epsilon]]=1)