Probability and Mathematical Statistics icon (the first chapter the basic concepts of probability theory)
1.2 Probability
1, the concept of
Probability: an objective measure of the size of the possibility of occurrence of random time
Frequency: Frequency ≠ probability, it can only be estimated as a probability
Classical probability: the limited nature, such as the possibility of
Geometric probability: classical probability of promotion, will "and other possibilities" to promote "uniformity"
2, axiomatic definition of probability
1. Non-Negative: 0≤P (A) ≤1
2. Normative: P (Ω) = 1
Example 3. can (fully) Additivity: Case mutually exclusive - events and event probability and probability =
3, the probability of nature
1) P (F) = 0
2) Limited additivity
3) Monotonicity
４）P（A）＝１－P（A）
5) general formula subtraction
- P（A）=P（B）- P（AB）
6) AB not mutually exclusive, the adder general formula
- P（A∪B）＝P（A）＋P（B）－P（AB）
- Mutually exclusive events probability = probability event to do together and
4, the probability calculation summary
Addition and subtraction operations and the corresponding probability event
Multiplied by the probability / conditional probability by event-related independence, not attributable to understand probability calculation Wayne map, and belongs to the conditional probability multiplication formula
5, the conditional probability P (A | B) = P (AB) / P (B)
Text of the statement: at the event B occurs, the probability of occurrence of event A condition
P (A | B) is not determined in relation to the size of P (A)
P (A | B)> P (A), B contributing to the occurrence of A
P (A | B) <P (A), B A is hindered occurred
P (A | B) = P (A), B has no influence on the occurrence of A
6, the multiplication formula P (AB) = P (B ) P (A | B)
Quadrature event probability
7, sample spatial correlation formula
Divide the sample space
Event satisfies incompatible, and collection and for the entire sample space, said event B1, B2, ......, Bn is a finite sample space S is divided
That pay is Φ, and that S? Probability is greater than 0
Total probability publicity of key application is to find divide
Icon:
P（B1）—P（A|B1）—>
P（B2）—P（A|B2）—>P(A)
P（Bn）—P（A|Bn）—>
Total probability formula P (A) = Σ (k = 1-> n) P (Bk) P (A | Bk)
- prove:
- A is included in S = ∪ (k = 1-> n) Bk
- A＝A∩S＝∪（ｋ＝１－>ｎ）（Bｋ×A）
- Addition Formula P (A) = Σ (k = 1-> n) P (Bk × A)
- Multiplication formula P (A) = Σ (k = 1-> n) P (Bk) P (A | Bk)
Conversion and Multiplication formula: P (AB) = P (A × 1) = P (A)
Applied by known demand forecast result, priori probability
Bayesian formula
A: So the probability of an event rather than the outcome of an event, Bi: all different reasons (division)
P (Bj | A): under A condition has occurred, Bj leads probability of its occurrence
P (ABj): A probability of occurrence of a cause Bj
P (A): the probability of occurrence of A / A probability of occurrence in the case of B1-Bn
1) If the request is applied to perform due to speculation, posterior probability
Prior probability: P (B1)
Posterior probability: P (B1 | A)
Total probability formula: prior probability P (Bi) * P (A | Bi) summation
Bayesian formula: posterior probability P (B1 | A)
2) understanding the Bayesian formula
In the next general after the total probability formula obtained by asking, as the denominator as a condition has occurred, it is a molecular formula of total probability, conditional probability.
Only conditional probability think there is no problem.
Example: the same problem in
In a total probability formula (1): P (B1) P (A | B1) prior to
(2) Molecular Bayes formula: P (B1 | A) is the post
8, the independence of events
1) the definition of the formula:
（1）P（Ａ｜Ｂ）=P（A）
( 2) P (AB) = P (A) P (B) radically
2) define the pairwise independent (three conditions):
P（AB）=P（A）P（B）
P（BC）=P（B）P（C）
P (AC) = P (A) P (100)
3) A, B, C each independently defined (four conditions):
P（AB）=P（A）P（B）
P（BC）=P（B）P（C）
P (AC) = P (A) P (100)
P（ABC）=P（A）P（B）P（C）
4) the inevitable event and the impossible event and other events independent of each other
A, B are independent, then A and non-B, non-A and B, and non A non B independently
9, standardized answers
1) set the probability of an event rather than a set
2) determination of probability, respectively
3) Write a formula name, the formula column: multiplication formula, total probability formula
When the answer to the symbol value and probability matching
Description The reset event is a division, are independent events (independent events)
10, typical questions / tips
1) If the results described ball taken before the k-th, calculated, the k th 1st probability =
2) P (A non-B non) = P (A concomitant B non) = 1-P (A concomitant B) = 1- [P (A) + P (B) -P (AB)]
3）P（A非B）＝P（B－AB）＝P（B）－P（AB）
4) the probability of an event and an expression P (A1∪A2) = P (A1) + P (A2) -P (A1A2)
5) n one cross-probability event expression
Under independent conditions, P = 1-P ((A1∪A2∪ ...... ∪An) non) = 1-P (A1 A2 non non non ...... An) = 1-P (A1 non) P (non-A2) ... ... P (An non)
6) pay the probability of an event expression
Under independent conditions, P (A1A2) = P (A1) P (A2)
7) Key examples
Ai sample space division - total probability formula
Ci independent events
8) P (three independent division event occurs only where a) = P (C1C2 C2C3 non-C3 non ∪C1 non non non non-C ∪C1 2C3) by mutually incompatible = P (C1C2 non non C3) + P (C1 non C2C3 non) + P (C1 non-C2 non-C3) = P (C1) P (C2 non) P (C3 non) + P (C1 non) P (C2) P (C3 non) + P (C1 non ) P (C2) P (C3 non)
11, the wrong questions
1) the probability of an event is not necessarily a sure thing?
Fall into a random point within a circle, the center point drop is less than the probability: 0, but may still occur
2）PPT1_3，P18