二项式系数的概念给人最直观的概念就是,这里有n个物品,分成两组,其中一组的数量是i的所有组合情况. 它的证明过程既可以从组合分析的角度,也可以从数学归纳的角度,由于数学归纳涉及到计算比较困难,我们这里就呈现二项式定理的数学归纳的证明方法. 定理证明的过程中用到了如下的二项式系数恒等关系,是基于递推求二项式系数的一个方法: 基于对二项式系数和二项式定理的理解,我们下面可将其推广到多项式系数和多项式定理.其证明过程是类似的.…

在概率论问题中求解基本事件.某个事件的可能情况数要涉及到组合分析. 而这一部分主要涉及到简单的计数原理和二项式定理.多项式定理. 我们从一个简单的实例入手. 方程的整数解个数: Tom喜欢钓鱼,一直他在r天中钓了n条鱼,设xi表示Tom第i天钓鱼的数目,这里我们,很显然时间是有序排列的,因此我们得到一个r元向量<x1,x2,x3……,xr>,那么满足上述条件,即x1+x2+x3+……+xr=n的r元组合.有多少个呢? 分析:首先我们刻意的将问题限制一下,假设每天Tom都不是空手而归,那么通过插…

[BZOJ2318]Spoj4060 game with probability Problem Description Alice和Bob在玩一个游戏.有n个石子在这里,Alice和Bob轮流投掷硬币,如果正面朝上,则从n个石子中取出一个石子,否则不做任何事.取到最后一颗石子的人胜利.Alice在投掷硬币时有p的概率投掷出他想投的一面,同样,Bob有q的概率投掷出他相投的一面. 现在Alice先手投掷硬币,假设他们都想赢得游戏,问你Alice胜利的概率为多少. Input 第一行一个正整数t,…

We start with the fuzzy binomial. Then we discuss the fuzzy Poisson probability mass function. Fuzzy Binomial Let $E$ be a non-empty, proper subset of $X=\{x_1,x_2,x_3,...,x_n\}$. Let $P(E)=p$ so that $P(E^{'})=1-p$ where $p\in (0,1)$. Suppose we hav…

Let $X=\{x_1,x_2,...,x_n\}$ be a finite set and let $P$ be a probability function defined on all subsets of $X$ with $P(\{x_i\})=a_i,~1\leq i \geq n,~0<a_i<1$ for i and $\sum^{n}_{i=1}=1$. $X$ together with $P$ is a discrete (finite) probability dis…

PDF version PDF & CDF The probability density function is $$f(x; \mu, \sigma) = {1\over\sqrt{2\pi}\sigma}e^{-{1\over2}{(x-\mu)^2\over\sigma^2}}$$ The cumulative distribution function is defined by $$F(x; \mu, \sigma) = \Phi\left({x-\mu\over\sigma}\ri…

PDF version PDF & CDF The probability density function of the uniform distribution is $$f(x; \alpha, \beta) = \begin{cases}{1\over\beta-\alpha} & \mbox{if}\ \alpha < x < \beta\\ 0 & \mbox{otherwise} \end{cases} $$ The cumulative distribu…

PDF version PDF & CDF The exponential probability density function (PDF) is $$f(x; \lambda) = \begin{cases}\lambda e^{-\lambda x} & x\geq0\\ 0 & x < 0 \end{cases}$$ The exponential cumulative distribution function (CDF) is $$F(x; \lambda) =…

PDF version PMF Suppose that a sample of size $n$ is to be chosen randomly (without replacement) from an urn containing $N$ balls, of which $m$ are white and $N-m$ are black. If we let $X$ denote the number of white balls selected, then $$f(x; N, m,…

PDF version PMF Suppose there is a sequence of independent Bernoulli trials, each trial having two potential outcomes called "success" and "failure". In each trial the probability of success is $p$ and of failure is $(1-p)$. We are obs…