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概率论与数理统计离散型随机变量(概率论与数理统计之随机变量与离散随机变量)

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随机变量取值的随机性,随机变量取到各个可能值具有一定的概率,且取到所有可能值的概率之和为1.

随机变量的分类

随机变量按其取值的离散情况分为两类:

1)离散型:所有可能值为有限个或可列个的。

2)非离散型:包括连续型的和其它的。

离散型随机变量的定义:

如果一个随机变量的所有可能取值为有限个或可列无穷多个,则这样的随机变量称为离散型随机变量。

离散型随机变量X的分布律或概率分布:

离散型随机变量X的所有可能取值的概率列:

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称为X的分布律或概率分布。

分布律的性质:

由概率的定义,P满足如下两个条件:

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(0-1)分布

随机变量X只可能取0与1两个值,它的分布律是

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Eg:抛一枚硬币观察得到正面或反面。若将硬币抛n次,就是n重伯努利试验。

服从(0-1)分布和二项分布随机变量的关系:它们描述的都是伯努利试验的随机变量。

区别在于:

(0-1)分布随机变量描述的是一次伯努利试验,而二项分布随机变量描述的是多次伯努利试验事件A发生的概率统计。因此,服从二项分布的随机变量X可以分解成n个(0-1)分布随机变量之和,即

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泊松分布

若随机变量X的所有可能取值为:0,1,2,...

而取各个可能值的概率为:

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历史上,泊松分布是为二项分布概率的近似计算,于1837年由法国数学家泊松引入的。

泊松定理:

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英文翻译

Random Variables and Discrete Random Variables

The randomness of the value of a random variable, the random variable has a certain probability of taking each possible value, and the sum of the probabilities of taking all possible values is 1.

Classification of random variables

Random variables are divided into two categories according to the discrete situation of their values:

1) Discrete type: all possible values are finite or can be listed.

2) Non-discrete type: including continuous type and others.

The definition of discrete random variable:

If all the possible values of a random variable are finite or infinitely numberable, then such a random variable is called a discrete random variable.

2. Bernoulli distribution, binomial distribution

Definition: Tests with only two possible results A and A are collectively called Bernoulli test. The Bernoulli test is independently repeated n times under the same conditions. This series of tests is called an n-fold Bernoulli test.

If X is the number of occurrences of event A in the n-fold Bernoulli test, p is the number of occurrences of event A in each test, and p is the probability of event A occurring in each test.

Then it is said that X obeys the binomial distribution with parameters n and p, denoted as X ~ B(n, p)

Eg: Toss a coin and observe the heads or tails. If the coin is tossed n times, it is an n-fold Bernoulli test.

3. The relationship between (0-1) distribution and binomial distribution random variables

They describe random variables in Bernoulli's experiment.

The difference is that:

(0-1) The distributed random variable describes a Bernoulli test, while the binomial distributed random variable describes the probability statistics of multiple Bernoulli test events. Therefore, the random variable X that obeys the binomial distribution can be decomposed into the sum of n (0-1) distributed random variables, namely

4. Historically, the Poisson distribution is an approximate calculation of the probability of the binomial distribution, introduced in 1837 by the French mathematician Poisson.

参考资料:百度

英文翻译:来源于Google翻译

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