QM4_Probability

时间:2021-07-15 01:19:36
QM4_Probability

  QM4_Probability

Basic Concepts

Probability concepts

Terms
  • Random variable
    • A quantity whose possible values are uncertain.
  • Outcomes
    • The possible values of a random variable.
  • Event
    • A specified set of outcomes.
Properties
  • 0 <= P(E) <=1
  • QM4_Probability

Events

QM4_Probability

Odds (赔率)

Odds for the event E
  • P(E)/[1-P(E)]
Odds against the event E
  • [1-P(E)]/P(E)
Example
  • Goven (horse will win the race) = 1/8, what are the odds for or against the horse will win the race?
    • Odds for horse will win the race = (1/8) / (1-1/8) = 1/7
    • Odds against horse will win the race = (1-1/8) / (1/8) = 7/1

Rule

Multplication rule **

  • P(A|B) = P(AB)/P(B)
  • P(AB) = P(B)xP(A|B) = P(A)xP(B|A)
  • For mutually exclusive enents: P(AB) = 0
  • For independent events: P(AB) = P(A)P(B)

Addition rule

  • P(A+B) = P(A)+P(B)-P(AB)
  • For mutually exclusive events: P(A+B) = P(A)+P(B)
  • QM4_Probability
  • 含义:
    • P(A|B) : 在B发生的条件下A发生;
    • P(B|A): 在A发生的条件下B发生;
    • P(AB): A发生且B也发生.

Total probability rule (全概率法则)

  • 全概率法则, 包含了所有可能发生的情况
  • Definition
    • explains the unconditional probability of the event in terms of probabilities conditional on the scenarios.
  • Formula
    • P(A) = P(A|S1)P(S1) + P(A|S2)P(S2)... + P(A|Sn)P(Sn)
    • where S1,S2...Sn are mutually exclusive and exhaustive.
    • 其实就是乘法法则 P(A|S1)P(S1) = P(AS1)
    • QM4_Probability
    • 其实就是乘法法则, P(AS1)+P(AS2)+P(AS3)+P(AS1=4)
      •   P(AS1)= PS1(PA|S1)

Baye's formula (贝叶斯公式)

  • Definition
    • given a prior probabilities P(A) for an event, if you receiv new information (B), the rule for updating your probability(posterior probability, P(A|B)) of the event.
  • forula:
  • QM4_Probability
  • 例;

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Probability Statistics

Expected value **

  • Definition
    • the probability-weighted average of the possible outcomes of the random variable(X)
  • Calculation
    • E(X) = P(X1)X1 + P(X2)X2 ... + P(Xn)Xn
        • QM4_Probability

          QM4_Probability

        • E(x)就是x拔, 求方差也就是在求期望. 期望中的权重变成了这里方差中的概率.

  • 其实求方差也是在求加权平均
  • 算期望就是算加权平均. 相当于基于概率的加权平均. 所以计算期望需要两个值, probability和value.
  • QM4_Probability

Covariance (协方差) ***

  • Definition
    • A easure of how two variables move together. 两个随机变量变动的方向性.
  • Calculation
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  • Characteristics
    • Positive covariance: the two variables tend to move together. 你涨我也涨,你跌我也跌.
    • Negative covariance: the two variables tend to move in apposite direction. 你涨我跌.
    • Valuses range from minus infinity to positive infinity
    • Units of covariance difficult to interpret (比如若是人的平方, 这样的单位没有任何意义)
    • Autocovariance is equal to the variance? 这句话怎么理解?
      • QM4_Probability
        • 协方差是衡量两个变量,所以把(x-x拔)(x-x拔)的其中一个x变成了y. 其实和方差是一样的. 

Correlation (相关系数,相关性) ***

  • Definition
    • A standardized measure of linear relationship between two variables.
  • Calculation
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      • 分母是标准差, 分子是协方差, 即协方差 / 标准差.

      • 例:
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        • 相关系数 = 协方差 / 标准差, 题目中协方差是0.80, 标准差没有告诉 , 告诉的是variance 0.0036/ 0.0009, 需要开个根号.
        • covariance = correlation * 标准差, 即, covariance = 0.80*0.03*0.06 = 0.00144.
  • Characteristics
    • Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation)
    • A correlation of 0 indicates an absence of any linear(straight-line) relationship and doesn't indicate independence (相关系数是0只说明两个变量没有线性关系, 不代表两个变量互不影响. 比如y=x*x,虽然不是线性关系, 但是抛物线)
    • The bigger the absolute value, the stronger the linear relationship.
    • QM4_Probability