Math/Probabilistic

1.2. Probabilistic Models

First-Penguin 2023. 8. 8. 19:33
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"A probabilistic model is a mathematical description of an uncertain situation."

Elements of a Probabilistic Model

  • The sample space \(\Omega\).
  • The probability law
    • an event \(A\) (a set of possible outcomes)
    • a probability of \(A\); \(P(A)\) (non-negative number)
    • that encodes one's knowledge or belief about the collective likelihood of the elements of \(A\).
    • The probability law must satisfy certain properties to be introduced shortly.

Figure 1.2: The main ingredients of a probabilistic model.

 

 

Sample Spaces and Events

  • The experiment is an underlying process that every probabilistic model involves, from which exactly one of possible outcomes occurs.
  • The sample space of the experiment : the set of all possible outcomes, and is denoted by \(\Omega\).
  • The event : a collection of possible outcomes

 

Choosing an Appropriate Sample Space

Mutually exclusive & collectively exhaustive

  • Regardless of their number, each element of the sample space should be distinct and mutually exclusive, so that when the experiment is carried out there is a unique outcome.
    • Mutually exclusive : One trial can only have one come at a time
  • The sample space chosen for a probabilistic model must be collectively exhaustive, in the sense that no matter what happens in the experiment, one always obtains an outcome that has been included in the sample space.
    • Collectively exhaustive : The number of 'all' cases must be taken into account
  • In addition, the sample space should have enough detail to distinguish between all outcomes of interest to the modeler, while avoiding irrelevant details.

 

 

 

Sequential Models

- sample space for a pair of rolls

- Tree-based sequential description

Figure 1.3: Two equivalent descriptions of the sample space of an experiment involving two rolls of a 4-sided die.

 

Probability Laws

The sample space \(\Omega\)  associated with an experiment.

The probability law assigns to every event \(A\) and the probability of \(A\) is a number \(P(A)\).

 

Probability Axioms

  1. (Non-negativity) \(P(A) \geq 0\), for every event \(A\).
  2. (Additivity) If \(A\) and \(B\) are two disjoint events, then the probability of their union satisfies

    $$ P(A \cup B) = P(A) + P(B). $$
    More generally, if the sample space has an infinite number of elements and \(A_1, A_2, \cdots\) is a sequence of disjoing events, then the probability of their union satisfies
    $$ P(A_1 \cup A_2 \cup \cdots) = P(A_1) + P(A_2) + \cdots. $$ 
  3. (Normalization) The probability of the entire sample space \(\Omega\) is equal to 1, that is, \(P(\Omega) = 1\)

* 3개 axioms로 P(A)<=1 derive 가능 -> 정리하기

 

Discrete Models

 

Discrete probability law

 

If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element.

In particular, the probability of any event \(\{s_1,s_2, \cdots, s_n\}\) is the sum of the probabilities of its elements:

 

$$P(\{s_1,s_2,\cdots,s_n\}) = P(s_1) + P(s_2) + \cdots + P(s_n).$$

 

 

Discrete uniform probability law

 

If the sample space consists of n possible outcomes which are equally likely (i.e. all single-element events have the same probability), then the probability of any event A is given by

 

$$P(A) = \frac{\text{number of elements of} \ A}{n}.$$

Continuous Models

 

Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterise the probability law.

 

 

Properties of Probability Laws

 

Consider a probability law, and let A, B, and C be events.

  • (a) If \(A \in B\), then \(P(A) \leq P(B).\)
  • (b) \(P(A \cup B) = P(A) + P(B) - P(A \cap B).\)
  • (c) \(P(A \cup B) \leq P(A) + P(B).\)
  • (d) \(P(A \cup B \cup C) = P(A) + P(A^{c} \cap B) + P(A^{c} \cap B^c \cap C).\)

 

Models and Reality

* Not everything in the textbook is included.

Figure 1.6: Visualization and verification of various properties of probability laws using Venn diagrams.

 

 

 

Bertrand's paradox

더보기

베르트랑의 역설(Bertrand's paradox)은 19세기 프랑스의 수학자 조세프 베르트랑(Joseph Bertrand)에 의해 제기된 확률 이론의 역설적인 문제입니다. 이 역설은 다양한 방식으로 정의될 수 있지만, 가장 잘 알려진 형태는 다음과 같습니다.

원의 둘레에 완전히 랜덤하게 한 점을 찍을 때, 그 점을 기준으로 만들어지는 무작위한 선분 중에서 어떤 선분이 원 내부를 완전히 둘러싸고 있는지의 확률을 생각해 보는 문제입니다. 이때 선분의 길이는 원의 둘레를 기준으로 랜덤하게 선택됩니다.

베르트랑의 역설은 선분의 길이를 선택하는 방법에 따라서 이 문제의 답이 다르게 나온다는 것을 보여줍니다. 선분의 길이를 선택하는 방법으로는 다음 세 가지가 가장 일반적으로 고려됩니다.

1. 원의 지름에 평행한 선분을 선택한다.
2. 원의 중심으로부터 일정 거리 이하의 선분을 선택한다.
3. 원의 둘레에서 임의의 두 점을 선택하고, 그 두 점을 연결하는 선분을 선택한다.

이 세 가지 방법에 따라 선분의 길이를 선택하면, 선분이 원 내부를 완전히 둘러싸고 있는 경우의 확률은 서로 다른 값으로 나타납니다. 이러한 다양한 결과로 인해 베르트랑의 역설은 확률 이론의 해석과 무작위성에 대한 이해를 깊이 있게 탐구하는 계기가 되었습니다. 이후 많은 수학자들이 이 문제에 대해 연구하고 다양한 해석을 제시하였습니다.

 

 

References

Bertsekas, D. P., Tsitsiklis, J. N. (2008). Introduction to Probability Second Edition. Athena Scientific.