Week 4 機率論

Basic Probability Concepts Random Trials

 

Sample Spaces

  

Events

1. Complementary Event

2. Joint Occurrence Event

3. Occurrence of One Event or Another

4. Mutually Exclusive Event

Probability Probability measure

P(E) denotes the probability that event E will occur

Probability Postulates （機率基本三公設）1.

2.

3.

Some Consequences of the Postulates

1. The sum of the probabilities of all basic outcomes in the sample spaces is 1

2. Every event has a probability value between 0 and 1

3. If E1 and E2 are two mutually exclusive events, then the probability of both occurring is zero.

Meaning of Probability 隨機試驗的特性 (1) 可重複執行的隨機試驗 (repeatable random expe

riment) 可以在完全相同的條件下反覆執行的隨機試驗

(2) 不可重複執行的隨機試驗 (unrepeatable random experiment) 無法在完全相同的條件下反覆執行的隨機試驗

Objective Interpretation Subjective Interpretation

Classical or a prior probability

Relative frequency or a posterior probability

Probability distribution

Univariate, Bivariate, and Multivariate probability distributions

Joint, Marginal, and Conditional Probability distributions

Relationships between variables Independent events

1.

2. 3.

Random variables A random variable is a variable whose

numerical value is determined by the outcome of a random trial.

1. Discrete random variables

2. Continuous random variables

Discrete random variables Probability distribution P(X=x) with each

distinct outcome x. Cumulative probability distribution

P(X<=x) Joint, Marginal, and Conditional probability

distributions

Expected value of random variable Variance of Random variable Standard Deviation of random variable Probability density function Cumulative probability function

Bernoulli Random Variable

Binomial Probability Function