Note15 RV

• 笔记

• RV

  • 定义： (Random Variable). A random variable X on a sample space Ω is a function X : Ω → R that assigns to each sample point ω ∈ Ω a real number X(ω).
  • 我们将限制我们的关注于离散随机变量，即它们在有限或可数无限的范围中取值，这意味着尽管我们定义X将Ω映射到R，但X实际取的值集{X(ω)：ω ∈ Ω}是R的离散子集

• Probability Distribution

  • 定义：The distribution of a discrete random variable X is the collection of values {(a,P[X = a]) : a ∈ A }, where A is the set of all possible values taken by X

    • X 为Ω中的每个可能的样本点分配一个唯一的概率值，总和恰好为1

  • Bernoulli Distribution

    • 两个可能值，一个p，一个1-p，参数为 p 的 bernoulli 分布
  • Binomial Distribution
    • n个可能值，从Set采样是independent的，参数为 n、p的Binomial 分布
  • Hypergeometric Distribution
    • n个可能值，从Set采样是dependent的，参数为 N、B、n的超几何分布

• Multiple RV和Independence

  • Joint Distribution定义：The joint distribution for two discrete random variables X and Y is the collection of values {((a,b),P[X = a,Y = b]) : a ∈ A , b ∈ B}, where A is the set of all possible values taken by X and B is the set of all possible values taken by Y .

    • 当给定X和Y的联合分布时，X的分布P[X = a]被称为边缘分布, 有\(P[X = a] = \sum_{b\in B} P[X = a,Y = b]\)

  • Independence定义：Random variables X and Y on the same probability space are said to be independent if the events X = a and Y = b are independent for all values a,b. Equivalently, the joint distribution of independent r.v.’s decomposes as P[X = a,Y = b] = P[X = a]P[Y = b], ∀a,b.

    • 如果\(I_i\)表示第i次掷硬币得到H的指示随机变量，那么\(I_1,...,I_n\)是相互独立的随机变量。这个例子激发了常用短语“独立同分布（i.i.d.）的随机变量集”。在这个例子中，\(\{I_1,...,I_n\}\)是一组独立同分布的指示随机变量集。 --- 同分布就是指各随机变量都是一样的概率分布，这里都是Berboulli Distribution
  • 对于随机变量而言，独立的含义是指，一个随机变量取任何可能值对另一个随机变量取值的概率无影响

• Expectation

  • 定义：The expectation of a discrete random variable X is defined as \(E[X] = \sum_{a\in A} a×P[X = a]\), where the sum is over all possible values taken by the r.v.

• Linearity of Expectation

  • 定理：For any two random variables X and Y on the same probability space, we have \(E[X +Y] = E[X] +E[Y].\) Also, for any constant c, we have\(E[cX] = cE[X].\)

  • 一些应用

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