Unit 1: Probability models and axioms

1. Lec. 1: Probability models and axioms

1.1. Sample Space

  • Definition: the set of all of possible outcomes is called the sample space of the experiment,. Denoted by Ω\Omega
  • A subset of the sample space is called event. Probability is assigned to events.
  • sample space can be finite, infinite, continuous, etc.
  • The construction of a sample space is a description of the possible outcomes of a probabilistic experiment.

Examples

  • discrete/finite example:
    • Two rolls of a tetrahedral die.
      • roll twice, get 16 outcomes. (the order sometimes matters)
  • continuous example:
    • (x, y) such that 0x,y10 \le x, y \le 1: a square box.

1.2. Probability axioms

  • Non-negativity: P(A)0P(A) \ge 0
  • Normalization: P(Ω)=1P(\Omega) = 1
  • (Finite) additivity: If AB=,then P(AB)=P(A)+P(B)\text{If } A \cap B = \emptyset,\text{then } P(A \cup B) = P(A) + P(B)
    • \emptyset denote empty set.
    • AB= A \cap B = \emptyset means A and B are disjoint events.

Consequences

  • P(A)1P(A) \le 1
  • P()=0P(\emptyset) = 0
  • P(A)+P(Ac)=1P(A) + P(A^c) = 1
  • P(ABC)=P(A)+P(B)+P(C)P(A \cup B \cup C) = P(A) + P(B) + P(C)
    • A, B and C are disjoint events.
    • and similarly for k disjoint events.
  • P(s1,s2,,sk)=P(s1s2sk)=P(s1)+P(s2)++P(sk)P({s_1, s_2, \ldots, s_k}) = P({s_1} \cup {s_2} \cup \ldots \cup {s_k})= P({s_1}) + P({s_2}) + \ldots + P({s_k})
  • If AB=ϕ,then P(A)P(B)\text{If } A \subset B = \phi,\text{then } P(A) \le P(B)
  • P(AB)=P(A)+P(B)P(AB)P(A \cup B) = P(A) + P(B) - P(A \cap B)
  • P(AB)P(A)+P(B)P(A \cup B) \le P(A) + P(B)
  • P(ABC)=P(A)+P(AcB)+P(AcBcC)P(A \cup B \cup C) = P(A) + P(A^c \cap B) + P(A^c \cap B^c \cap C)

1.3. Discrete Models

  • Example: Coin tosses. Consider an experiment involving a single coin toss. There are two possible outcomes, heads (H) and tails (T). The sample space is Ω={H,T}\Omega = \{H, T\}, and the events are {H,T},{H},{T},.\{H, T\}, \{H\}, \{T\}, \emptyset.

Discrete Probability Law

  • the probability of any event {s1,s2,,sn}\{s_1, s_2, \ldots, s_n \} is the sum of the probabilities of its elements: P({s1,s2,,sn})=P({s1})+P({s2})++P({sn})P(\{s_1, s_2, \ldots, s_n \}) = P(\{s_1\}) + P(\{s_2\}) + \ldots + P(\{s_n\})

Discrete Uniform Law

  • Assume Ω\Omega consists of n equals likely elements
  • Then the probability of any event A is given by P(A)=Number of elements ofAnP(A) = \frac{\text{Number of elements of} A}{n}

1.4. Continuous Models

  • Example: (x, y) such that 0x,y10 \le x, y \le 1
  • P((x,y)x+y1/2)=121212=18P({(x, y)| x + y \le 1/2}) = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}
    • Because it's a right triangle with length 1/2 of each side.
  • P((0.5,0.3))=0P({(0.5, 0.3)}) = 0
    • Because it's the area of a single point.

1.5. Countable additivity

Probability calculation: discrete but infinite sample space

  • Keep tossing a coin and the outcome is the number of tosses until we observe heads for the first time.
  • Probability: P(n)=12n, n=1,2,P(n) = \frac{1}{2^n},\ n = 1,2,\ldots
  • Legitimate check: P(Ω)=1P(\Omega) = 1
    • n=112n=12n=012n=12111/2=1\sum_{n = 1}^{\infty} \frac{1}{2^n} = \frac{1}{2}\sum_{n = 0}^{\infty} \frac{1}{2^n} = \frac{1}{2} \cdot \frac{1}{1-1/2} = 1
      • Geometric Series Theorem
  • P(outcome is even)=P({2,4,6,})=P({2}),P({4}),P({6}),})=122+124+126+=141114=13P(\text{outcome is even}) = P(\{2, 4, 6, \ldots\}) = P(\{2\}), P(\{4\}), P(\{6\}), \ldots\}) = \frac{1}{2^2} + \frac{1}{2^4} + \frac{1}{2^6} + \cdots = \frac{1}{4} \cdot \frac{1}{1 - \frac{1}{4} } = \frac{1}{3}
    • To support this calculation, we'll need:

Countable Additivity Axiom

  • If A1,A2,A_1, A_2, \ldots is an infinite sequence of disjoint events, then P(A1A2)=P(A1)+P(A2)+P(A_1 \cup A_2 \cup \cdots) = P(A_1) + P(A_2) + \cdots
  • Notice the events should be sequence, like the example above.
    • Example (x, y) such that 0x,y10 \le x, y \le 1 can not be arranged in a sequence. and the elements are NOT countable.
  • Additivity holds only for "countable" sequences of events.

Countable and uncountable sets

  • Countable: can be put in 1-1 correspondence with positive integers
    • positive integers: 1,2,3,1, 2, 3, \ldots
    • integers: 0,1,1,2,2,0, 1, -1, 2, -2, \ldots
    • pairs of positive integers
    • rational numbers q with 0 < q < 1
      • 1/2,1/3,2/3,1/4,3/4,1/5,2/5,1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, \ldots
  • Uncountable:
    • the interval [0, 1]
    • the reals, the plane...

1.6. Interpretations of probability theory

  • A framework for analyzing phenomena with uncertain outcomes
    • Rules for consistent reasoning
    • Used for predictions and decisions

2. Words

  • tetrahedral [,tetrə'hi:drəl, -'he-] adj. 四面体的;有四面的
  • axiom ['æksiəm] n. [数] 公理;格言;自明之理
  • discrete [dis'kri:t] adj. 离散的,不连续的
  • legitimate [li'dʒitimət, li'dʒitimeit] adj. 合法的;正当的;合理的;正统的 vt. 使合法;认为正当(等于legitimize)
  • additivity [,ædi'tivəti] n. 添加;相加性
  • unit square [数] 单位平方形

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