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Random Processes: Mean and Variance
(m10656)
Author:
Michael Haag
Keywords:
average
,
DSP
,
mean
,
random
,
random process
,
random signal
,
random signals
,
variance
Summary:
(Blank Abstract)
Subject:
Mathematics and Statistics,
Science and Technology
Language:
English
Popularity:
99.62%
Revised:
20050405
Revisions:
4
Random Selection
(m23652)
Author:
Paul E Pfeiffer
Keywords:
Basic sequence
,
Compound demand
,
Counting random variable
,
Counting random variables
,
Incremental sequence
,
Matlab and compound demand
,
Random sums
Summary:
The usual treatments deal with a single random variable or a fixed, finite number of random variables, considered jointly. However, there are many common applications in which we select at random a member of a class of random variables and observe its value, or select a random number of random ... various situations.
[Expand Summary]
The usual treatments deal with a single random variable or a fixed, finite number of random variables, considered jointly. However, there are many common applications in which we select at random a member of a class of random variables and observe its value, or select a random number of random variables and obtain some function of those selected. This is formulated with the aid of a counting or selecting random variable N, which is nonegative, integer valued. It may be independent of the class selected, or may be related in some sequential way to members of the class. We consider only the independent case. Many important problems require optional random variables, sometimes called Markov times. These involve more theory than we develop in this treatment. As a basic model, we consider the sum of a random number of members of an iid class. In order to have a concrete interpretation to help visualize the formal patterns, we think of the demand of a random number of customers. We suppose the number of customers N is independent of the individual demands. We formulate a model to be used for a variety of applications. Under standard independence conditions, we obtain expressions for compound demand D, conditional expectation for g(D) given N = n, and moment generating function for D. These are applied in various situations.
[Collapse Summary]
Subject:
Mathematics and Statistics
Language:
English
Popularity:
48.85%
Revised:
20090918
Revisions:
6
Random Signals
(m10989)
Author:
Nick Kingsbury
Keywords:
random signals
Summary:
This module introduces random signals.
Subject:
Mathematics and Statistics
Language:
English
Popularity:
93.17%
Revised:
20050607
Revisions:
6
Random Signals and Noise
(m31819)
Author:
Phil Schniter
Keywords:
autocorrelation
,
expectation
,
filtering
,
noise
,
power spectral density
,
power spectrum
,
PSD
,
random process
,
random signal
,
uncorrelated
,
white noise
,
widesense stationary
,
zeromean
Summary:
This module describes the basics of random signals and noise as needed for an introductory course on data communications. First it introduces the notion of power spectral density (PSD) and defines "white" noise as a noise with a flat PSD. Next it describes the effect that linear filtering has on ... fundamental properties.
[Expand Summary]
This module describes the basics of random signals and noise as needed for an introductory course on data communications. First it introduces the notion of power spectral density (PSD) and defines "white" noise as a noise with a flat PSD. Next it describes the effect that linear filtering has on a random signal, in both time and frequency domains. For this, the autocorrelation function is introduced, and expectations are used to derive the fundamental properties.
[Collapse Summary]
Subject:
Mathematics and Statistics
Language:
English
Popularity:
57.48%
Revised:
20090916
Revisions:
3
Random Variables and Probabilities
(m23260)
Author:
Paul E Pfeiffer
Keywords:
applied probability
,
mapping
,
probability
,
random variables,
Summary:
Often, each outcome of an experiment is characterized by a number. If the outcome is observed as a physical quantity, the size of that quantity (in prescribed units) is the entity actually observed. In many nonnumerical cases, it is convenient to assign a number to each outcome. For example, in ... probability analysis.
[Expand Summary]
Often, each outcome of an experiment is characterized by a number. If the outcome is observed as a physical quantity, the size of that quantity (in prescribed units) is the entity actually observed. In many nonnumerical cases, it is convenient to assign a number to each outcome. For example, in a coin flipping experiment, a “head” may be represented by a 1 and a “tail” by a 0. In a Bernoulli trial, a success may be represented by a 1 and a failure by a 0. In a sequence of trials, we may be interested in the number of successes in a sequence of n component trials. One could assign a distinct number to each card in a deck of playing cards. Observations of the result of selecting a card could be recorded in terms of individual numbers. In each case, the associated number becomes a property of the outcome. The fundamental idea of a real random variable is the assignment of a real number to each elementary outcome ω in the basic space Ω. Such an assignment amounts to determining a function X, whose domain is Ω and whose range is a subset of the real line R. Each ω is mapped into exactly one value t, although several ω may have the same image point. Except in special cases, we cannot write a formula for a random variable X. However, random variables share some important general properties of functions which play an essential role in determining their usefulness. Associated with a function X as a mapping are the inverse mapping and the inverse images it produces. By the inverse image of a set of real numbers M under the mapping X, we mean the set of all those ω∈Ω which are mapped into M by X. If X does not take a value in M, the inverse image is the empty set (impossible event). If M includes the range of X, (the set of all possible values of X), the inverse image is the entire basic space Ω. The class of inverse images of the Borel sets on the real line play an essential role in probability analysis.
[Collapse Summary]
Subject:
Mathematics and Statistics
Language:
English
Popularity:
66.02%
Revised:
20090918
Revisions:
9
Random Variables and Probability Density Functions
(m11246)
Author:
Don Johnson
Subject:
Mathematics and Statistics,
Science and Technology
Language:
English
Popularity:
89.38%
Revised:
20030801
Revisions:
2
Random Vectors
(m11249)
Author:
Don Johnson
Subject:
Mathematics and Statistics,
Science and Technology
Language:
English
Popularity:
79.17%
Revised:
20030808
Revisions:
2
Random Vectors
(m10988)
Author:
Nick Kingsbury
Keywords:
random vectors
Summary:
This module introduces random vectors.
Subject:
Mathematics and Statistics
Language:
English
Popularity:
79.86%
Revised:
20050607
Revisions:
6
Random Vectors and Joint Distributions
(m23318)
Author:
Paul E Pfeiffer
Keywords:
applied probability
,
joint distribution
,
random variables
,
random vectors
Summary:
Often we have more than one random variable. Each can be considered separately, but usually they have some probabilistic ties which must be taken into account when they are considered jointly. We treat the joint case by considering the individual random variables as coordinates of a random vector. We extend ... joint distribution.
[Expand Summary]
Often we have more than one random variable. Each can be considered separately, but usually they have some probabilistic ties which must be taken into account when they are considered jointly. We treat the joint case by considering the individual random variables as coordinates of a random vector. We extend the techniques for a single random variable to the multidimensional case. To simplify exposition and to keep calculations manageable, we consider a pair of random variables as coordinates of a twodimensional random vector. The concepts and results extend directly to any finite number of random variables considered jointly. If the joint distribution for a random vector is known, then the distribution for each of the component random variables may be determined. These are known as marginal distributions. In general, the converse is not true. However, if the component random variables form an independent pair, the treatment in that case shows that the marginals determine the joint distribution.
[Collapse Summary]
Subject:
Mathematics and Statistics
Language:
English
Popularity:
80.05%
Revised:
20090918
Revisions:
8
Random Vectors and MATLAB
(m23320)
Author:
Paul E Pfeiffer
Keywords:
applied probability
,
matlab
,
mprocedures
,
probability
,
simple random variables
Summary:
The systematic formulation in the previous module Minterms shows that each Boolean combination, as a union of minterms, can be designated by a vector of zeroone coefficients. A coefficient one in the ith position (numbering from zero) indicates the inclusion of minterm Mi in the union. We formulate this ... and solution.
[Expand Summary]
The systematic formulation in the previous module Minterms shows that each Boolean combination, as a union of minterms, can be designated by a vector of zeroone coefficients. A coefficient one in the ith position (numbering from zero) indicates the inclusion of minterm Mi in the union. We formulate this pattern carefully below and show how MATLAB logical operations may be utilized in problem setup and solution.
[Collapse Summary]
Subject:
Mathematics and Statistics
Language:
English
Popularity:
88.15%
Revised:
20090918
Revisions:
7
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