Stochastic models - time series.

Slides:

Advertisements

Similar presentations

Random Processes Introduction (2)

Advertisements

STATISTICS Joint and Conditional Distributions

The Normal Distribution

Discrete Probability Distributions

Probability Theory STAT 312 STAT 312 Dr. Zakeia AlSaiary.

Probability theory 2010 Order statistics  Distribution of order variables (and extremes)  Joint distribution of order variables (and extremes)

1 Def: Let and be random variables of the discrete type with the joint p.m.f. on the space S. (1) is called the mean of (2) is called the variance of (3)

Time Series Basics Fin250f: Lecture 3.1 Fall 2005 Reading: Taylor, chapter

Probability Distributions

4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of.

Standard Normal Distribution

I.1 ii.2 iii.3 iv.4 1+1=. i.1 ii.2 iii.3 iv.4 1+1=

Data Basics. Data Matrix Many datasets can be represented as a data matrix. Rows corresponding to entities Columns represents attributes. N: size of the.

2. Random variables  Introduction  Distribution of a random variable  Distribution function properties  Discrete random variables  Point mass  Discrete.

I.1 ii.2 iii.3 iv.4 1+1=. i.1 ii.2 iii.3 iv.4 1+1=

Continuous Random Variables and Probability Distributions

Discrete Random Variables and Probability Distributions

Approximations to Probability Distributions: Limit Theorems.

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

The Multivariate Normal Distribution, Part 2 BMTRY 726 1/14/2014.

Multivariate Probability Distributions. Multivariate Random Variables In many settings, we are interested in 2 or more characteristics observed in experiments.

Joint Probability distribution

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

Joint Probability Distributions

Sampling Distributions  A statistic is random in value … it changes from sample to sample.  The probability distribution of a statistic is called a sampling.

Review of Probability.

On Rearrangements of Fourier Series Mark Lewko TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAA A AAA A A A A.

Limits and the Law of Large Numbers Lecture XIII.

Estimating parameters in a statistical model Likelihood and Maximum likelihood estimation Bayesian point estimates Maximum a posteriori point.

LECTURE IV Random Variables and Probability Distributions I.

Sampling and sampling distibutions. Sampling from a finite and an infinite population Simple random sample (finite population) – Population size N, sample.

Lecture PowerPoint Slides Basic Practice of Statistics 7 th Edition.

Tim Marks, Dept. of Computer Science and Engineering Random Variables and Random Vectors Tim Marks University of California San Diego.

Stats Probability Theory Summary. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes.

Random processes. Matlab What is a random process?

Chapter 3 DeGroot & Schervish. Functions of a Random Variable the distribution of some function of X suppose X is the rate at which customers are served.

The generalization of Bayes for continuous densities is that we have some density f(y|  ) where y and  are vectors of data and parameters with  being.

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

Stochastic models - time series. Random process. an infinite collection of consistent distributions probabilities exist Random function. a family of random.

Random Variables. Numerical Outcomes Consider associating a numerical value with each sample point in a sample space. (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

Random Variables A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment. A random variable can be.

1 EE571 PART 3 Random Processes Huseyin Bilgekul Eeng571 Probability and astochastic Processes Department of Electrical and Electronic Engineering Eastern.

Central Limit Theorem Let X 1, X 2, …, X n be n independent, identically distributed random variables with mean  and standard deviation . For large n:

Summary of the Last Lecture This is our second lecture. In our first lecture, we discussed The vector spaces briefly and proved some basic inequalities.

Unit 4 Review. Starter Write the characteristics of the binomial setting. What is the difference between the binomial setting and the geometric setting?

APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS

Math a Discrete Random Variables

Probability Theory and Parameter Estimation I

Main topics in the course on probability theory

Chapter 5 Joint Probability Distributions and Random Samples

Sample Mean Distributions

The distribution function F(x)

Lecture 09: Gaussian Processes

Chapter 7: Sampling Distributions

Econ Roadmap Focus Midterm 1 Focus Midterm 2 Focus Final Intro

AP Statistics: Chapter 7

Stochastic time series.

Stochastic models - time series.

ASV Chapters 1 - Sample Spaces and Probabilities

Lecture Slides Elementary Statistics Twelfth Edition

Lecture Slides Elementary Statistics Twelfth Edition

Stochastic models - time series.

STOCHASTIC HYDROLOGY Random Processes

E370 Statistical analysis for bus & econ

Stochastic models - time series.

Lecture 10: Gaussian Processes

Central Limit Theorem: Sampling Distribution.

Bernoulli Trials Two Possible Outcomes Trials are independent.

Random Variables A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment. A random variable can be.

Presentation transcript:

Stochastic models - time series. Random process. an infinite collection of consistent distributions probabilities exist Random function. family of random variables, {Y(t;), t  Z, } Z = {0,±1,±2,...},  a sample space

Specified if given F(y1,...,yn;t1 ,...,tn ) = Prob{Y(t1)y1,...,Y(tn )yn } n = 1,2,... F's are symmetric in the sense F(y;t) = F(y;t),  a permutation F's are compatible F(y1 ,...,ym ,,...,;t1,...,tm,tm+1,...,tn} = F(y1,...,ym;t1,...,tm) m+1  t = 2,3,...

Finite dimensional distributions First-order F(y;t) = Prob{Y(t)  t} Second-order F(y1,y2;t1,t2) = Prob{Y(t1)  y1 and Y(t2)  y2} and so on

Normal process/series. Finite dimension distributions multivariate normal Multivatiate normal. Entries linear combinations of i.i.d standard normals Y =  +  Z : s by 1 : s by r Y: s by 1 Z: Nr(0,I) I: r by r identity E(Y) =  var(Y) = ' s by s Conditional marginals linear in Y2 when condition on it

Other methods i) Y(t;), : random variable ii) urn model iii) probability on function space iv) analytic formula Y(t) =  cos(t + ) , : fixed : uniform on (-,]

There may be densities The Y(t) may be discrete, angles, proportions, vectors, ... Kolmogorov extension theorem. To specify a stochastic process give the distribution of any finite subset {Y(1),...,Y(n)} in a consistent way,  in A