概率论期中总结

Chapter 1 Introduction to Probability

1.3 Experiments, Events and Sample Space

? Types of Experiments ? The Sample Space The collection of all possible outcomes of an experiment is called the sample space of the experiment.

1.4 The Definition of Probability

? Axiom and Basic Theorems A?S, Pr(A) indicates the probability that Awill occur. A1. For every event A, Pr(A)?0. A2. Pr(S) = 1.

A3. For every infinite sequence of disjoint events A1, A2, . ..

??Pr(?Ai)?i?1?Pr(A)

ii?1Theorem ? Pr(?) = 0.

? For every finite sequence of n disjoint events A1, A2, . . .,

?Pr(?Ai)?i?1?Pr(A)

ii?1?? Theorem1.4.3. For every event A, Pr(Ac) = 1 ? Pr(A). ? If A?B, then Pr(A)?Pr(B). ? For every event A, 0?Pr(A)?1.

? For every two events A and B, Pr(A?B) = Pr(A) + Pr(B) ? Pr(AB).

1.5 Finite Sample Spaces

? Requirements of Probabilities s1, . . . , sn?S

p1, . . . , pn satisfy: the following two conditions: pi ?0 for i = 1, . . . , n and ?i?1nPi=1

The probability of each event A can then be found by adding the probabilities pi of all outcomes si that belong to A.

? Simple Sample Spaces A sample space S containing n outcomes s1, . . . , sn is called a simple sample space if the probability assigned to each of the outcomes s1, . . . , sn is 1/n. If an event A in this simple sample space contains exactly m outcomes, then Pr(A) =

mn.

1.6 Combinatorial Methods

Binomial Coefficients:

?n????k?=

n!k!?n?k?!

(Binomial theorem) For all numbers x and y and each positive integer n,

(x?y)nn?n?nn?k????xykk?0??

1.7 The Probability of a Union of Events

? For every three events A1, A2, and A3,

Pr(A1?A2?A3) = Pr(A1) + Pr(A2) + Pr(A3) ? [Pr(A1A2) + Pr(A2A3) + Pr(A1A3)] + Pr(A1A2A3). ? For every n events A1, . . . , An,

nPr(?Ai)?i?1

n?Pr(A)??Pr(AA)??iiji?1i?ji?j?kPr(AiAjAk)??i?j?k?lPr(AiAjAkAl)?(?1)n?1Pr(AiAj...An)Chapter 2 Conditional Probability

2.1 The Definition of Conditional Probability

? Introduction to the Definition The conditional probability of the event A given that the event B has occurred.: Pr(A | B).

Pr(A|B)?Pr(AB)Pr(B)

Rewrite the formula of conditional probabilities: Pr(AB) = Pr(B)Pr(A | B). Pr(AB) = Pr(A)Pr(B | A). ? Intersection of n events Suppose that A1,A2, . . . ,An are events such that Pr(A1A2 · · ·An?1) > 0. Then

Pr(A1A2 · · ·An) = Pr(A1)Pr(A2 | A1)Pr(A3 | A1A2) … Pr(An| A1A2 · · ·An?1). Rewrite:

Suppose that A1,A2, . . . ,An,B are events such that Pr(A1A2 · · ·An?1 | B) > 0. Then

Pr(A1A2 · · ·An | B) = Pr(A1 | B)Pr(A2 | A1B) · · · Pr(An | A1A2 · · ·An?1B).

2.2 Independent Events

? Definition of Independence Suppose that Pr(A) > 0 and Pr(B) > 0. Events A and B are independent if Pr(A | B) = Pr(A), Pr(B | A) = Pr(B).

Theorem ? Theorem ? If two events A and B are independent, then the events A and Bc are also independent, that is, Pr(ABc ) = Pr(A)Pr(Bc ).

? Let A1, . . . ,Ak be events such that Pr(A1 · · ·Ak ) > 0. Then A1, . . . ,Ak are independent if and only if, for every two disjoint subsets {i1, . . . , im} and {j1, . . . , jl} of {1, . . . , k}, we have Pr(Ai1 · · ·Aim | Aj1 · · ·Ajl) = Pr(Ai1 · · ·Aim).

2.3 Bayes’ Theorem

?

Theorem ? Suppose that the events B1, . . . ,Bk form a partition of the space S and that Pr(Bj ) > 0 for j = 1, . . . , k. Then, for every event A in S,

kPr(A)??Pr(B)Pr(A|B)

jjj?1Theorem ? (Conditional Version of Law of Total Probability.) The law of total probability has an analog conditional on another event C, namely

kPr(A|C)??Pr(Bj?1j|C)Pr(A|BjC)

Theorem (Bayes’ theorem) ? Let the events B1, . . . ,Bk form a partition of the space S such that Pr(Bj ) > 0 for j = 1, . . . , k, and let A be an event such that Pr(A) > 0.

Pr(Bj|A)?Pr(Bi)Pr(A|Bi)kThen, for i = 1, . . . , k,

?j?1Pr(Bj)Pr(A|Bj)

Chapter 3 Random Variables and

Distributions

3.1 Random Variables and Discrete Distributions ? Definition of a Random Variable. Consider an experiment for which the sample space is denoted by S. A real-valued function that is defined on the space S is called a random variable. In other words, in a particular experiment a random variable X would be some function that assigns a real number X(s) to each possible outcome s 2 S.

? Measure the random variables Preparation for the distribution of a random variable. Random variable X. A subset of the real line A, X?A denote some event. Determine the

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