'@GSMC/홍대형: Statistical Communication Theory'에 해당되는 글 11건
- 2009.06.11 term project
- 2009.05.27 Chapter 10 Stochastic Processes
- 2009.05.27 Chapter 9 Estimation of a Random Variable
- 2009.05.27 Chapter 8 Hypothesis Testing
- 2009.05.14 Chapter 7 Parameter Estimation Using the Sample Mean
- 2009.05.06 Chapter 6 Sums of Random Variables
- 2009.04.02 Homework #3
- 2009.04.01 Chapter 3 Continuous Random Variables
- 2009.03.17 Chapter 2. Discrete Random Variables
- 2009.03.03 Chapter 1. Experiments, Models, and Probabilities
- 2009.03.02 [Yates & Goodman] Probability and Stochastic Processes
Probability and Stochastic Process, 2nd ed.
Chapter 10 Stochastic Processes
http://en.wikipedia.org/wiki/Stochastic_process
One approach to stochastic processes treats them as functions of one or several deterministic arguments ("inputs", in most cases regarded as "time") whose values ("outputs") are random variables: non-deterministic (single) quantities which have certain probability distributions.
-> time suquence of the events
time structure of a process vs. amplitude structure of a random variable
(autocorrelation function and autocovariance function vs. expected value and variance)
Poisson
Brownian
Gaussian
Wide sense stationary processes
cross-correlation
10.1 Definitions and Examples
NRZ 파형
http://en.wikipedia.org/wiki/Non-return-to-zero
'@GSMC > 홍대형: Statistical Communication Theory' 카테고리의 다른 글
| term project (0) | 2009.06.11 |
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| Chapter 9 Estimation of a Random Variable (0) | 2009.05.27 |
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| Chapter 7 Parameter Estimation Using the Sample Mean (0) | 2009.05.14 |
| Chapter 6 Sums of Random Variables (0) | 2009.05.06 |
Probability and Stochastic Process, 2nd ed.
Chapter 9 Estimation of a Random Variable
prediction
A predictor uses random variables produced in early subexperiments to estimate a random variable produced by a future subexperiment.
9.1 Optimum Estimation Given Another Random Variable
The estimate of X that produces the minimum mean square error is the expected value (or conditional expected value) of X calculated with the probability model that incorporates the available information.
Bind estimation of X
Estimation of X given an event
Minimum Mean Square Estimation of X given Y
9.2 Linear Estimation of X given Y
9.3 MAP and ML Estimation
9.4 Linear Estimation of Random Varaiables from Random Vectors
'@GSMC > 홍대형: Statistical Communication Theory' 카테고리의 다른 글
| term project (0) | 2009.06.11 |
|---|---|
| Chapter 10 Stochastic Processes (0) | 2009.05.27 |
| Chapter 8 Hypothesis Testing (0) | 2009.05.27 |
| Chapter 7 Parameter Estimation Using the Sample Mean (0) | 2009.05.14 |
| Chapter 6 Sums of Random Variables (0) | 2009.05.06 |
Yates & Goodman
Probability and Stochastic Process, 2nd ed.
Chapter 8 Hypothesis Testing
statistical inference method
1) perform an experiment
2) observe an outcome
3) state a conclusion
http://en.wikipedia.org/wiki/Statistical_inference
8.1 Significance Testing
http://en.wikipedia.org/wiki/Significance_testing
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. "A statistically significant difference" simply means there is statistical evidence that there is a difference.
http://en.wikipedia.org/wiki/Hypothesis_testing
8.2 Binary Hypothesis Testing
http://en.wikipedia.org/wiki/Binary_classification
Maximum A Posteriori Probability (MAP) Test
http://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation
Minimum Cost Test
Neyman-Pearson Test
http://en.wikipedia.org/wiki/Neyman%E2%80%93Pearson_lemma
Maximum Likelihood Test
http://en.wikipedia.org/wiki/Maximum_likelihood
8.3 Multiple Hypothesis Test
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| Chapter 9 Estimation of a Random Variable (0) | 2009.05.27 |
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| Chapter 6 Sums of Random Variables (0) | 2009.05.06 |
| Homework #3 (0) | 2009.04.02 |
Chapter 7 Parameter Estimation Using the Sample Mean
statistical inference
http://en.wikipedia.org/wiki/Statistical_inference
Statistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population
7.1 Sample Mean: Expected Value and Variance
The sample mean converges to a constant as the number of repetitions of an experiment increases.
Althouth the result of a single experiment is unpredictable, predictable patterns emerge as we collect more and more data.
sample mean
= numerical average of the observations
: the sum of the sample values divided by the number of trials
7.2 Deviation of a Random Variable from the Expected Value
Markov Inequality
: an upper bound on thte probability that a sample value of a nonnegative random variable exceeds the expected value by any arbitrary factor
http://en.wikipedia.org/wiki/Markov_inequality
Chebyshev Inequality
: The probability of a large deviation from the mean is inversely proportional to the square of the deviation
http://en.wikipedia.org/wiki/Chebyshev_inequality
7.3 Point Estimates of Model Parameters
http://en.wikipedia.org/wiki/Estimation_theory
estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.
http://en.wikipedia.org/wiki/Point_estimation
the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" for an unknown (fixed or random) population parameter
relative frequency (of an event)
point estimates :
bias
consistency
accuracy
consistent estimator
: sequence of estimates which converges in probability to a parameter of the probability model.
The sample mean is an unbiased, consistent estimator of the expected value of a random variable.
The sample variance is a biased estimate of the variance of a random variable.
mean square error
: expected squared difference between an estimate and the estimated parameter
The standard error of the estimate of the expected value converges to zero as n grows without bound.
http://en.wikipedia.org/wiki/Law_of_large_numbers
7.4 Confidence Intervals
accuracy of estimate
confidence interval
: difference between a random variable and its expected value
confidence coefficient
: probability that a sample value of the random variable will be within the confidence interval
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| Homework #3 (0) | 2009.04.02 |
| Chapter 3 Continuous Random Variables (0) | 2009.04.01 |
http://en.wikipedia.org/wiki/Central_limit_theorem
6.1 Expected Values of Sums
Theorem 6.1
The expected value of the sum equals the sum of the expected values whether or not each variables are independent.
Theorem 6.2
The variance of the sum is the sum of all the elements of the covariance matirx.
6.2 PDF of the Sum of Two Random Variables
Theorem 6.5
http://en.wikipedia.org/wiki/Convolution
linear system
http://en.wikipedia.org/wiki/Linear_system
6.3 Moment Generating Functions
In linear system theory, convolution in the time domain corresponds to multiplication in the frequency domain with time functions and frequency functions related by the Fourier transform.
In probability theory, we can use transform methods to replace the convolution of PDFs by multiplication of transforms.
moment generating function
: the transform of a PDF or a PMF
http://en.wikipedia.org/wiki/Moment_generating_function
http://en.wikipedia.org/wiki/Laplace_transform
region of convergence
6.4 MGF of the Sum of Independent Random Variables
Theorem 6.8
Moment generating functions provide a convenient way to study the properties of sums of independent finite discrete random variables.
Theorem 6.9
The sum of independent Poisson random variables is a Poisson random variable.
Theorem 6.10
The sum of independent Gaussian random variables is a Gaussian random variable.
In general, the sum of independent random variables in one family is a different kind of random variable.
Theorem 6.11
The Erlang random variable is the sum of n independent exponential random variables.
6.5 Random Sums of Independent Random Variables
random sum
: sum of iid random variables in which the number of terms in the sum is also a random variable
It is possible to express the probability model of R as a formula for the moment generating function.
The number of terms in the random sum cannot depend on the actual values of the terms in the sum.
6.6 Central Limit Theorem
Probability theory provides us with tools for interpreting observed data.
bell-shaped curve = normal distribution
So many practical phenomena produce data that can be modeled as Gaussian random variables.
http://en.wikipedia.org/wiki/Central_limit_theorem
Central Limit Theorem
The CDF of a sum of random variables more and more resembles a Gaussian CDF as the number of terms in the sum increases.
Central Limit Theorem Approximation = Gaussian approximation
6.7 Applications of the Central Limit Theorem
De Moivre-Laplace Formula
http://en.wikipedia.org/wiki/Theorem_of_de_Moivre%E2%80%93Laplace
normal approximation to the binomial distribution
6.8 The Chernoff Bound
Chernoff Bound
http://en.wikipedia.org/wiki/Chernoff_bound
exponentially decreasing bounds on tail distributions of sums of independent random variables
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Yates & Goodman
Probability and Stochastic Process, 2nd ed.
3.1 The Cumulative Distribution Function
3.2 Probability Density Function
3.3 Expected Values
3.4 Families of Continuous Random Variables
http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)
http://en.wikipedia.org/wiki/Exponential_random_variable
http://en.wikipedia.org/wiki/Erlang_random_variable
3.5 Gaussian Random Variables
http://en.wikipedia.org/wiki/Gaussian_random_variable
3.6 Delta Functions, Mixed Random Variables
3.7 Probability Models of Derived Random Variables
3.8 Conditioning a Continuous Random Variable
3.9 MATLAB
http://en.wikipedia.org/wiki/Quantile_function
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2.2 Probability Mass Function
2.3 Families of Discrete Random Variables
http://en.wikipedia.org/wiki/Bernoulli_random_variable
2.4 Cumulative Distribution Function (CDF)
2.5 Averages
http://www.amstat.org/publications/jse/v13n2/vonhippel.html
http://en.wikipedia.org/wiki/Mode_(statistics)
2.6 Functions of a Random Variable
2.7 Expected Value of a Derived Random Variable
2.8 Variance and Stand Deviation
2.9 Conditional Probaility Mass Function
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2nd Ed. John Wiley & Sons, 2005
http://www.wiley.com/college/yates
출판사 페이지: http://he-cda.wiley.com/WileyCDA/HigherEdTitle/productCd-0471272140.html
Dr. Roy Yates received the B.S.E. degree in 1983 from Princeton University, and the S.M. and Ph.D. degrees in 1986 and 1990 from M.I.T., all in Electrical Engineering. Since 1990, he has been with the Wireless Information Networks Laboratory (WINLAB) and the ECE department at Rutgers, University. He is currently an associate professor.
David J. Goodman is Director of WINLAB and a Professor of Electrical and Computer Engineering at Rutgers University. Before coming to Rutgers, he enjoyed a twenty year research career at Bell Labs where he was a Department Head in Communications Systems Research. He has made fundamental contributions to digital signal processing, speech coding, and wireless information networks.
학습 페이지: http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0471272140&bcsId=1991
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