The first chapter of random events and probability

The first chapter of random events and probability

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Probability theory and mathematical statistics study is a random phenomenon:

• Probability theory: model study of random phenomena (probability distribution);
• Mathematical statistics: data collection and processing of random phenomena.

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§1.1 random events and operations

Random phenomenon phenomenon is not always appear the same result under certain conditions;:

Randomized trial : to be repeated under the same conditions random phenomenon observed is recorded, the experiment;

Sample space : a set consisting of all possible results substantially random phenomena; \ (\ Omega = \ {\ Omega \} \) , \ (\ Omega \) : Basic Results, also known as sample points ;

• Elements of the sample space may or may not be the number of the number;
• Sample space has at least two sample points, sample space comprises only two points of samples is the simplest sample space;
• Number of samples from the space containing the sample point to distinguish:
  • Discrete sample space: the number of sample points of a finite or countable;
  • Continuous sample space: the number of sample points is not an unlimited number of columns.

Random events : a set consisting of some random phenomena sample point, referred to the event;

• Is any event corresponding to a subset of the sample space; (Venn diagram);

• When a subset of a sample point there, said the event occurred;

• Events can represent a collection, you can also use unmistakable language to describe;

  The basic events	A subset of the sample space Ω individual elements thereof;
  Inevitable event	The largest subset of the sample space
  Unlikely event	Minimal subset of the sample space Ω, namely the empty set

Random variables : the variables used to represent the result of random phenomena, represents random variables should be stated meanings;

The relationship between events

• Inclusion relations: event A occurs inevitably lead to the occurrence of an event B;
• Equal relationship;
• Incompatible: A, B is not the same sample point.

Operation between events : union, intersection, difference, remainder.

• $ A \ cup B $: New Event A, B points in all samples composed of at least one of two events occur;

• \ (A \ CAP B \) : New Event A, B in a common sample dots;

\ (\ cup_ {i = 1 } ^ n A_i, \ cap_ {i = 1} ^ \ infty A_i \) cross and operation can be generalized to the case of infinite;

• \ (A \ setminus B \) : a new event in A and not the sample point event B is composed of;
  \ [\ {X-= A \} = \ {X-\ Leq A \} - \ {X-<A \}, \ {a <A \ leq b \} = \ {X \ leq b \} - \ {X \ leq a \} \]

• \ (\ overline {A} \) : opposite event;
  \ [A \ setminus B = A \ B ^ CAP C \]

Operational nature of the event :

• Commutative: A \ (\ CAP \) B = B \ (\ CAP \) A
• 结合律：\((A\cup B)\cup C=A\cup(B\cup C)\)
• Distributive law: \ ((A \ B Cup) \ CAP C = the AC \ Cup the BC \)
• De Morgen公式：\(\overline{\cup_{i=1}^\infty A_i}=\cup_{i=1}^\infty \overline{A_i}\)

Event domain

• A sample space and a certain subset of the set of calculation results consisting of classes, referred to as \ (F. \) , The event field to have closure to collection operation, and:

  And post operation can be realized by opposition;
  difference may be achieved by calculating the cross opposition;

• In this way, and the opposition is the most basic operation, then define the event field is as follows:

  Set \ ([Omega] \) is present as space, \ (F. \) Of \ ([Omega] \) collection class consisting of some subset of, if \ (F. \) Is satisfied:

  • \(Ω∈F\)；
  • If \ (a∈F \) , the opposite event \ (\ overline {A} ∈ F \) ;
  • If \ (A_n∈F, n-1,2 ... .. = \) , and belonging to the column may be \ (F. \) .

  Called \ (F \) is a field event, also known as \ (\ sigma \) domain or \ (\ sigma \) Algebra.

• In probability theory, also known as \ ((\ Omega, F) \) is a measurable space.

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Definition and method for determining the probability of §1.2

Axiomatic definition of probability \ (Kolmogrov \)

Set \ (\ Omega \) is a sample space, \ (F \) as \ (\ Omega \) a field event of certain subsets of, if for any event \ (A \ in F \) , defined in \ (F. \) a real-valued function on \ (P (a) \) is satisfied:

• Theorem nonnegative if \ (A \ in F. \) , Then \ (P (A) \ GEQ 0 \) ,

• Axiom of regularity \ (P (\ Omega). 1 = \) ,

• Countable additivity \ (if A_1, A_2, \ cdots, A_n , \ cdots incompatible, then: P (\ cup_ {i = 1} ^ \ infty A_i) = \ sum_ {i = 1} ^ \ infty (A_i) \) ,

Called \ (P (A) \) for the event A probability , said the three elements \ ((\ Omega, F, P) \) is a probability space

Determine the probability of frequency method

In a large number of replicate experiments, with a stable frequency values ​​to obtain a probability

• A study events associated with a large number of random phenomena can be repeated;

• In n replicate experiments, denoted \ (n (A) \) is the number of event A occurs, also known as \ (n (A) \) of the event A is the frequency
  \ [f_n (A) = \ frac {n ( A)} {n} \]

• The frequency of occurrence of event A

• With the increase of the number of repetitions n of the experiment, the frequency \ (f_n (A) \) will be stabilized at a constant a, the constant is called a stable value of the frequency .

Determine the probability of the classical method

• Random phenomena involved only a limited number of sample points;

• Each sample point is equal to the possibility of occurrence;

• If event A contains sample points k, the probability of an event A is
  \ [P (a) = \ frac { the number of sample points contained in the event A} {\ the number of all sampling points Omega} = \ frac kn \]

In the classical approach, find the probability of an event A is attributed to calculate the number of sample points contained in A and \ (\ Omega \) the total number of sample points contained.

1. Sampling model
2. Sampling with replacement
3. Box model
4. Birthday problem

Geometric method to determine the probability of

Subjective method for determining the probability of