4. Random variables

【README】

This article summarizes from 《 Basic course of probability theory 》 by M.Ross , Wall crack recommendation ;

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【4.1】 A random variable

1） A random variable ：

• A real valued function defined on the sample space ; That is, random variable is a function （ Or a mapping , Map events to values ）;
• Let random variables X Is the sum of two dice points , We are concerned that the sum of points is 7, And don't care about the actual results (1,6) still (2,5).
• perhaps , Put random variables X Understood as a containing X Result events （ Events are subsets ）;

2） Cumulative distribution function F(x) （ It's very important *）

Be careful ： The cumulative distribution function is the distribution function .

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 【4.2】 Discrete random variables

1） Definition ：

• If a random variable has at most countable values , Then the random variable is a discrete random variable ;

2） Discrete random variables X The distribution column of ：p(a)   （ It's very important *）

 3） Distribution function F By distributing columns p(a) Calculate （ It's very important *）

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【4.3】 expect

1） Definition ：

• A random variable X My expectation is X A weighted average of all possible values , The weight of each value is X The probability of taking this value ;

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 【4.4】 The expectation of a function of random variables

1） Definition ：

• A random variable X, Random variable function   g(X) ,E[g(X)]  be called Random variable function g(X) The expectations of the ;

2） Calculate the expectation of random variable function
proposition 4.1

【 example 】 proposition 4.1 Calculate expectations

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  【4.5】 variance （ The deviation degree of random variable value relative to the mean ）

1） Definition ：

Add ： Discrete random variables are usually classified according to their distribution columns , Several common types of random variables are as follows .

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【4.6】 Bernoulli random variable and binomial random variable

1） Definition

 2） Binomial random variables X The nature of

 3） Calculate binomial distribution function

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【4.7】 Poisson random variable

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【4.8】 Other discrete probability distributions

【4.8.1】 Geometric random variable

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【4.8.2】 Negative binomial random variable

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【4.8.3】 Hypergeometric random variables

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【4.9】 Expectation of sum of random variables

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【4.10】 Properties of distribution function

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【 Summary 】 Probability distribution of discrete random variables , Expectation and variance