厦门大学：《概率论与数理统计 Probability and Statistics for Economists》课程教学资源（课件讲稿）Chapter 04 Important Probability Distributions

城险究鳞清粮南王季大門厦 Important Probability Distributions Professor Yongmiao Hong Cornell University June25,2019

Important Probability Distributions Professor Yongmiao Hong Cornell University June 25, 2019

CONTENTS 4.1 Introduction 4.2 Discrete Probability Distributions 4.3 Continuous Probability Distributions 4.4 Conclusion Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 2/135

Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 2/135 4.1 Introduction 4.2 Discrete Probability Distributions 4.3 Continuous Probability Distributions 4.4 Conclusion CONTENTS

Important Probability Distributions Introduction Introduction .A probability space (S,B,P)completely characterizes a random experiment. It is usually assumed that there exists a unique true but unknown probability distribution for the random exper- iment of interest. One often considers a class of probability distributions, say PMF/PDF f(x,0),indexed by parameter 0,where the functional form f(,)is known Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 3/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 3/135 Introduction Introduction

Important Probability Distributions Introduction Introduction Different values of 0 yield different distributions.The collection of these distributions constitute a family of probability distributions. This family of probability distributions is called a class of parametric probability distribution models. One main objective of statistics and econometrics is to use observed economic data to estimate the unknown true parameter value,say 00,under the assumption that there exists some parameter value 0o such that f(x,0o) coincides with the true probability distribution fx(x)of the random experiment,namely f(x,00)=fx(x). Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 4/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 4/135 Introduction Introduction

Important Probability Distributions Introduction Introduction This chapter introduces a variety of important distribu- tion models,and discuss their properties and applica- tions in economics and finance. We emphasize that it is important to t understand the meanings and roles that parameters play in each para- metric distribution. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 5/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 5/135 Introduction Introduction

CONTENTS 4.1 Introduction 4.2 Discrete Probability Distributions 4.3 Continuous Probability Distributions 4,4 Conclusion Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 6/135

Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 6/135 4.1 Introduction 4.2 Discrete Probability Distributions 4.3 Continuous Probability Distributions 4.4 Conclusion CONTENTS

Important Probability Distributions Discrete Probability Distributions Bernoulli Distribution A DRV X follows a Bernoulli(p)distribution if its PMF x={-p fx=1, if x=0, =p2(1-p)1-for x=0,1, 0.8 where 0<p<1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Remark: 0 Bernoulli(0.75) This is a binary random variable,taking value 1 with probability p,and taking value 0 with probability 1-p. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 7/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 7/135 Bernoulli Distribution Discrete Probability Distributions Remark:

Important Probability Distributions Discrete Probability Distributions Bernoulli Distribution Remarks: ●For Bernoulli(p), EX) P, var(X) p(1-p). ●MGF of Bernoulli(p) Mx(t)=pe +1-p for -oo<t<oo. Because p=P(X 1)and X only takes two possible values,parameter p fully characterizes the probability distribution of Bernoulli(p). Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 8/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 8/135 Bernoulli Distribution Discrete Probability Distributions Remarks:

Important Probability Distributions Discrete Probability Distributions Bernoulli Distribution Remarks: Bernoulli(p)has wide applications in e economics and finance: Neftci (1984)examines the asymmetry of U.S. business cycles by considering the time series properties of signed processes from a posi- tive change to a negative change,and from a negative change to a positive change. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 9/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 9/135 Bernoulli Distribution Discrete Probability Distributions Remarks:

Important Probability Distributions Discrete Probability Distributions Bernoulli Distribution Remarks: -Das (2002)approximates jumps of interest rates by a sequence of Bernoulli random variables. -Logit regression model P(X=1Z)= 1 1+exp(-B'Z) Important applications of logit regression include pattern recognition and classification analysis. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 10/135

Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 10/135 Bernoulli Distribution Discrete Probability Distributions Remarks: