What is a Random Variable?
Ans: "In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon."
In probability theory, "a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers"
https://en.wikipedia.org/wiki/Random_variable
What Is the Central Limit Theorem (CLT)?
"In the study of probability theory, the central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution (also known as a “bell curve”), as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population distribution shape."
https://www.investopedia.com/terms/c/central_limit_theorem.asp
"The Central Limit Theorem and Means
An essential component of the Central Limit Theorem is that the average of your sample means will be the population mean. In other words, add up the means from all of your samples, find the average and that average will be your actual population mean. Similarly, if you find the average of all of the standard deviations in your sample, you’ll find the actual standard deviation for your population. It’s a pretty useful phenomenon that can help accurately predict characteristics of a population. Watch a video explaining this phenomenon, or read more about it here: The Mean of the Sampling Distribution of the Mean." https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/normal-distributions/central-limit-theorem-definition-examples/
What is stochastic behavior?
"The word "stochastic" means "pertaining to chance" (Greek roots), and is thus used to describe subjects that contain some element of random or stochastic behavior. For a system to be stochastic, one or more parts of the system has randomness associated with it. https://www.cds.caltech.edu › courses › cds101 › faq › 02-10-07_stochastic
What is the meaning of stochastic process?
"A stochastic process means that one has a system for which there are observations at certain times, and that the outcome, that is, the observed value at each time is a random variable. Stochastic Processes – an overview | ScienceDirect Topics"
https://www.sciencedirect.com › topics › neuroscience › stochastic-processes
Stationary ergodic process
"In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.""
What is a stationary process?
"Stationarity is the property of a random process which guarantees that its statistical properties, such as the mean value, its moments and variance, will not change over time. A stationary process is one whose probability distribution is the same at all times. For more information see stationary process."
Several sub-types of stationarity are defined: first-order, second-order, nth-order, wide-sense and strict-sense. For details please see the reference above.
What is a Ergodic process
"An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average."" https://en.wikipedia.org/wiki/Stationary_ergodic_process
Ergodic Process:
"In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. … Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate."
https://en.wikipedia.org › wiki › Ergodic_process
A correlation function
"A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables."
https://en.wikipedia.org › wiki › Correlation_function
What is energy spectral density?
"Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing; that is, the energy of a signal is. Spectral density – Wikipedia"
https://en.m.wikipedia.org › wiki › Spectral_density
Spectral density estimation
"In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal.[1] Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities."
https://en.wikipedia.org/wiki/Spectral_density_estimation
Markov Process — from Wolfram MathWorld
mathworld.wolfram.com › … › Interactive Demonstrations
"Markov Process. A random process whose future probabilities are determined by its most recent values."
Poisson Process
"A Poisson Process is a model for a series of discrete event where the average time between events is known, but the exact timing of events is random. The arrival of an event is independent of the event before (waiting time between events is memoryless).""
Poisson Distribution
"The Poisson Process is the model we use for describing randomly occurring events and by itself, isn’t that useful. We need the Poisson Distribution to do interesting things like finding the probability of a number of events in a time period or finding the probability of waiting some time until the next event." https://towardsdatascience.com/the-poisson-distribution-and-poisson-process-explained-4e2cb17d459
Statistical signal processing
"Statistical signal processing is an approach which treats signals as stochastic processes, utilizing their statistical properties to perform signal processing tasks. Statistical techniques are widely used in signal processing applications.""
https://en.wikipedia.org › wiki › Signal_processing
Linear MMSE Estimation
https://www.probabilitycourse.com/chapter9/9_1_6_linear_MMSE_estimat_of_random_vars.php
Harmonic analysis
"Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis)."
https://en.wikipedia.org › wiki › Harmonic_analysis
By
Sayed Ahmed
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