概率论快速学习05：随机变量 二项分布 泊松分布

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“不要在意这些细节。”生活就是这样,每天过的好一点.

Written In The Font

• 随机变量

• 二项分布

• 泊松分布

• 泊松分布与二项分布的关系

Content

Random variable

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Random variable is usually understood to mean a real-valued random variable; this discussion assumes real values. A random variable is a real-valued function defined on a set of possible outcomes, the sample space Ω. That is, the random variable is a function that maps from its domain, the sample space Ω, to its range, the real numbers or a subset of the real numbers. It is typically some kind of a property or measurement on the random outcome.

                        

what is it? I think the definition is not clear to understand.In my words,Random variable can be used to describe the process of rolling dice and the possible outcomes.

Discrete probability distribution

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The range of distribution function is the discrete random variables, such as the onlyinteger is belongs to the discrete distribution. represents the probability random variable value. If the value of X is,then:

                      

Example:

若罚球两次, 第一次罚中的概率为0.75，若第一次罚中则第二次罚中的概率为0.8，若第一次未罚中则第二次罚中的概率为0.7.以X记罚球两次其中罚中的次数，求X的分布律。

解：X的可能取值为0，1，2.

X	0	1	2
pk	0.075	0.325	0.6

也可以通过一系列数据展现到图上

                     

Binomial Distribution

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the binomial distribution is the discrete probability distributionof the number of successes in a sequence of n independent yes/no experiments.In general, if the random variable X follows the binomial distribution with parameters n and p, we writeX ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function:

                      

Symbols for:

                     X～B (n, p)

show an example:

若某人做某事的成功率为1%，他重复努力 400次，则至少成功一次的概率为 :

                 

爱迪生: “天才＝1%的灵感＋99%的汗水” 但那1%的灵感是最重要的，甚至比那99%的汗水都要重要

Poisson's distribution

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A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if, for k=0,1,2,…, the probability mass function of X is given by:

                

Symbols for:

                 

The relation of binomial distribution and poisson's distribution

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when n –> ∞ , p < 0.1 :
                     

proved:

                   

Editor's Note

有些人不需要你去生气,做好你自己.自己开心过好每一天,这样就足够了.