计量经济学2.

2.4 PROBABILITY
Rules of probability:
1) If A, B, C,...are any events, they are said to be statistically independent events if: P(ABC...)=P(A)P(B)P(C)P(…) 2) If events A, B, C, ... are not mutually exclusive, then P(A+B)=P(A)+P(B)－P(AB)
2.4 PROBABILITY
1．The Classical or A Priori Definition： If an experiment can result in n mutually
exclusive and equally likely outcomes, and if m of these outcomes are favorable to event A, then P(A), the probability that A occurs, is m/n
P ( A) m n t he num ber of out com es favorablet o A t he t ot alnum ber of out com es
Two features of the probability: （1）The outcomes must be mutually exclusive; （2）Each outcome must have an equal chance of occurring.
2.4 PROBABILITY
3. Properties of probabilities
(1) 0≤P(A)≤1 (2) If A, B, C, ... are mutually exclusive events, then: P(A+B+C+...)=P(A)+P(B)+P(C)+... (3) If A, B, C, ... are mutually exclusive and collectively exhaustive set of events, P(A+B+C+...)=P(A)+P(B)+P(C)+...=1
Single/Univariate probability distribution functions Multivariate probability distribution functions Two-variate/Bivariate PDF
Joint probability: the prob. that the r.v. X takes a
2. Marginal Probability Density Function
——The relationship between the univariate PDFs, f(X)
or f(Y), and the bivariate joint PDF:
The marginal probability of X: the probability that X
2. PDF of a Continuous Random Variable
The probability for a continuous r.v. is always measured over an interval For a continuous r.v. the probability that such an r.v. takes a particular numerical value is always zero.
given value and Y takes a given value. Bivariate/joint PDF～f(X, Y) （1）Discrete joint PDF: f(X, Y) =P(X=x, Y=y) =0 when X≠x,Y≠y
（2）Continuous joint PDF: （omitted）
2.Relative Frequency or Empirical Definition
Frequency distribution: how an r.v. are distributed.
Absolute frequencies: the number of occurrence of a given event. Relative frequencies: the absolute frequencies divided by the total number of occurrence.

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2.2 EXPERIMENT, SAMPLE SPACE, SAMPLE POINT, AND EVENTS 1. Experiment A statistical/random experiment: a process
leading to at least two possible outcomes with uncertainty as to which will occur.
2.Sample space or population The population or sample space: the set of
assumes a given value regardless of the values taken by Y.
all possible outcomes of an experiment
3. Sample Point
Sample Point : each member, or outcome, of the sample space (or population)
4. Events

An event: a collection of the possible outcomes of an experiment; that is, it is a subset of the sample space. Mutually exclusive events: the occurrence of one event prevents the occurrence of another event at the same time. Equally likely events: one event is as likely to occur as the other event. Collectively exhaustive events: events that exhaust all possible outcomes of an experiment
1. A discrete random variable—— an r.v. that takes on only a finite (or accountably infinite) number of values. 2. A continuous random variable——an r.v. that can take on any value in some interval of values.
Empirical Definition of Probability: if in n trials（or observations）,m of them are favorable to event A, then P（A）, the probability of event A, is simply the ration m/n,（that is, relative frequency） provided n, the number of trials, is sufficiently large In this definition, we do not need to insist that the outcome be mutually exclusive and equally likely.
3. Cumulative Distribution Function (CDF)
F(X)=P(X ≤x) P(X ≤x): the probability that the r.v. X takes a value of less than or equal to x, where x is given. In fact, a CDF is merely an “accumulation” or simply the sum of the PDF for the values of X less than or equal to a given x.
DISTRIBUTION FUNCTION (PDF)
The probability distribution function or probability density function (PDF) of a random variable X: the values taken by that random variable and their associated probabilities. 1.PDF of a Discrete Random Variable Discrete r.v.(X) takes only a finite(or countably infinite) number of values. Probability distribution or probability density function (PDF)—it shows how the probabilities are spread over or distributed over the various values of the random variable X. PDF of a discrete r.v.(X) f(X)= P(X=Xi) for i=1,2,3...,n 0 for X≠Xi
CHAPTER 2
A REVIEW OF BASIC STATISTICAL CONCEPTS
2.1 SOME NOTATION
1. The Summation Notation
X
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X
can be abbreviated as: X or 2. Properties of the Summation k nk Operator
Conditional probability of A, given B
P( A | B) P ( A B) P( B)
Conditional probability of B, given A
P ( B | A) P ( A B) P ( A)
2.5 RANDOM VARIABLES AND PROBABILITY
CDF of a discrete r.v.: step function; CDF of a continuous r.v.: continuous curve
2.6 MULTIVARIATE PROBABILITY DENSITY FUNCTIONS
1. Two-variate/Bivariate PDF
2.3 RANDOM VARIABLES
A random/stochastic variable（r.v., for short): a
variable whose (numerical) value is determined by the outcome of an experiment.