MTH-102复习1

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随机变量之前的内容

Lec1

  1. Permutations

    1. 排列指的是在n个物品中选择k个物品进行排序，那么有多少种可能。
    2. 例如从5个字母中选择3个进行组词，即为
    3. 故

  2. Combinations

    1. 组合指的是在n个物品中选择k个物品，那么有多少种可能。
    2. 例如从5个字母中随机选择3个候选，即为
    3. 为选择3个进行排列的结果，但排列考虑了3个字母的顺序，组合中顺序不必被考虑，那么只需要将顺序除去，即
    4. 故
    5. 排列也可使用如下表达

  3. First step of probability

    1. Probability of counting: Example 1

      1. Problem: Compute the probability of obtaining exactly three '6's after rolling a fair dice 4 times.
      2. Solution:
        将一个骰子扔四次，共有种可能
        结果有4个位置，其中3个必须是6，那么选择这3个位置的方式为。剩下一个位置有五种可能性，那么刚好拥有3个6的可能有
        因此，结果是

    2. Probability of counting: Example 2

      1. Problem: Given a class of 6 girls and 5 boys.
        • (a)
          What is the probability that a committee of five, chosen at random from the class, consists of 3 girls and 2 boys?
        • (b)
          What is the probability that a committee of five, chosen at random from the class, consists of at least one boy?
      2. Solution:
        • (a)
          从11个人中选择5个人组成一组的方式有
          从6个女生中选择3个和从5个男生中选择2个的方式分别为因此，选择一个包括3个女生和2个男生的组的方式有
          因此，6个女生和5个男生组成刚好有3个女生和2个男生的组的概率为
        • (b)
          自a得，11人种选5人组成一组有462种方式
          至少有1男生，即男生数量可能为1，2，3，4，5。即男生数量为0的可能性不存在。
          男生数量为0则有种可能。
          那么，概率为

Lec2

  1. Set theory

    1. Random experiment

      The outcome of an experiment is not predictable with certainty
      随机实验指的是实验结果无法被确定预测的实验

    2. Sample space

      The set of all possible outcomes of an experiment
      实验的所有可能的实验结果的集合

    3. Events

      Any subset of the sample space, i.e. a set consisting of possible outcomes of the experiment
      样本空间的任意子集。即一个包括了实验的可能结果的集合

      1. Events: logical relations 事件的逻辑联系

        • 假设样本空间为S， E、F为两个事件
        1. 当实验结果包含了E，那么我们称E已发生(E has occurred)
        2. the union of E and F, i.e. either E or F occur. 表示事件E、F至少发生了一个
        3. the intersection of E and F, i.e. both E and F occur. 表示事件E、F同时发生
        4. the complement of E, i.e. E does not occur. 表示事件E没有发生
        5. E is contained in F, i.e. if E occurs, then F occurs. 事件E被包含在事件F中，即事件E发生时，事件F必然发生。
        6. The null event the event consisting no outcomes. 空集事件中没有任何实验结果
        8. if , then E and F are said to be mutually exclusive. E交F为空时，指EF是相互排斥的
        9. and are mutually exclusive
        10. if are events, thendenotes the union of these events, i.e. at least one of these events occurs.
        11. if are events, thendenotes the intersection of these events, i.e. all these events occur

    4. Rules from the set theory

      Let E, F, G be the subsets of S, then the following are satisfied.

      1. Commutative laws:

      2. Associative laws:

      3. Distributive laws:

      4. DeMorgan's laws:

      • 当以的形式出现时，运算优先级更高。即：

    5. Example

      例如一个投掷硬币的随机实验，他的样本空间即为，H为正面，T为反面
      当E={H}，那么E就是硬币为正面的事件

  2. Axioms of probability

    1. Basic

      • Consider an experiment whose sample space is S. For each event E of S, we assume that a number P(E) is defined and satisfies the following three axioms:
        3. For any sequence of mutually exclusive events (that is, events for which when ),
      • We refer to P(E) as the probability of the event E.

    2. Basic propositions of probability

      1. Complementation rule

        For any event E,

      2. Total probability law

        For any event A and B, it holds that

      3. Addition rule

        For any event A and B, it holds that

Lec3

  1. Conditional probability

    • 指在B事件发生的情况下A事件发生的概率
    • 当A、B都为样本空间S的事件，并且

  2. The multiplication rule

    • If A and B are events of a sample space S with and , then
    • The general case:
      If are events of a sample space S with for then

  3. Law of total probability

    • For any event A and B, it holds that
    • The general case:
      Let the events be mutually exclusive and
      We assume that for i = 1,2,...,n. Then for any event A, A can be represented as the union of mutually exclusive events, i.e.
      Therefore,

Lec4

  1. Bayes' rule

    Let the events be mutually exclusive with and
    For any event A, it follows by law of total probability that
    therefore for i = 1,2,...,n
    which is called Bayes' Rule

    被称为先验概率(Prior probability)，
    被称为后验概率(Posterior probability)，
    被称为似然(likelihood)

  2. Independent events

    Two events A and B are said to be independent if
    Two events A and B that are not independent are said to be dependent

    Remark: A and B are independent does not mean that
    Property: If A and B are independent, thenThe probability of A does not depend on the occurrence or nonoccurrence of B, and conversely