联合分布，条件分布的特点
Features of Joint and Conditional Distributions
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Random variables and their probability distributions
A random variable is one that takes on numerical values and has an outcome that is determined by an experiment.
Event:
• Simple: 掷一次骰子时面上的点数 • Compound: 掷两次骰子时面上的点数
Random variable: link the event with real line, so that for the simple event, random variable X can take a value in the set {1,2,3,4,5,6}. A particular realization x=5.
Population
Probability
Inferential Statistics
Sample
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为什么先温习概率再温习统计？
在概率相关的问题中，我们往往假定我们了解总体的特 性而对样本的一些具体问题给出回答。
在统计相关的问题中，我们了解样本的信息而期望对总 体特性加以推断。
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Flipping a Coin in History
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Random variables and their probability distributions: Discrete Random Variable
A discrete random variable is one that takes on only a finite number of values.
Event: a subset of outcomes contained in the sample space.
Random variable maps information into real line, and need to ensure that there is no information loss in the process.
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为什么先温习概率再温习统计？
要能够理解一个具体的样本究竟能够传递关于总 体的什么样的信息，我们首先需要理解从一个给 定的总体中抽取样本时究竟有什么样的不确定性。 不确定性在这世界无处不在，概率论提供了量化 描述这些不确定性的工具。
Before we can understand what a particular sample can tell us about the
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Random variables and their probability distributions: Discrete Random Variable
The probability density function (pdf) of X summarizes the information concerning the possible outcomes of X and the corresponding probabilities: f (X=xj) = pj, j=1,…,k, f (X≠xj) =0.
中级计量经济学
INTERMEDIATE ECONOMETRICS
温故而知新之概率论
北京大学中国经济研究中心 沈艳
为什么先温习概率再温习统计？
Population versus sample
Population: a well-defined collection of objects Sample: a subset of population
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Continuous Random variable: P(a<X<b)= P(a<X≤b)= P(a ≤X<b)
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Joint Distributions, Conditional Distributions and Independence
population, we should first understand the uncertainty associated with taking a sample from a given population.
Most aspects of the world around us have an element of randomness. The
Probability of an outcome is the proportion of time that the outcome occurs in the long run. It is used to describe the behavior of random variables.
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Random variables and their probability distributions: Binary Random Variable
A Bernoulli (binary) random variable
A random variable that only takes on values of 0 and 1. Example: the gender of the next student you meet on
Let fX(x) be the pdf of X and fY(y) be the pdf of Y. Then X and Y are independent iff fX,Y(x, y)= fX(x) fY(y). or P(X=x, Y=y)=P(X=x)P(Y=y).
Independence means knowing the outcome of X does not change the probabilities of the possible outcomes of Y and vice versa.
Knowing the pdf of a discrete random variable, one can calculate the probability of events evolving that random variable. e.g. Let X be the years spent at PKU of the next undergraduate you meet on campus, suppose P(X=1)= P(X=2)= P(X=3)= P(X=4)= 0.25, then P(X≥3)=0.5.
We have just described the behavior of a r.v., but we are usually interested in the occurrence of events involving more than one r.v..
Let X and Y be discrete random variables. Then (X,Y) has a joint distribution.
Any discrete random variable is completely described by listing its possible values and the associated probability that it takes on each value.
If X takes k possible values x1, x2,…xk, then the probabilities p1, p2,…pk , are defined by pj=P(X=xj), j=1,…,k p0+ p1 +… + pk =1
Random Variables and Their Probability Distribution
联合分布，条件分布和独立
Joint Distributions, Conditional Distributions, and Independence
概率分布的特点
Features of Probability Distributions
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Random variables and their probability distributions: Continuous Random Variable
A variable is a continuous random variable if it takes on any real value with zero probability.
theory of probability provide mathematical tools for quantifying and
describing this randomness.
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ቤተ መጻሕፍቲ ባይዱ
课程内容简介
对本课程中涉及到的概率方面的知识做基 本回顾。
随机变量和他们的概率分布
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Random variables and their probability distributions
Example:
experiment: 掷一次骰子
Outcomes:1 or 2 or 3 or 4 or 5 or 6