There are various ways to specify a random variable. The most visual is the probability density function (plot at the top), which represents how likely each value of the random variable is. The cumulative density function is a conceptually cleaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the moments, the cumulants, the characteristic function, the moment-generating function, and the cumulant-generating function. Some of these are very useful for the ...

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Normal distribution, Normal distribution - Overview, Normal distribution - History, Normal distribution - Specification of the normal distribution, Normal distribution - Probability density function, Normal distribution - Cumulative distribution function, Normal distribution - Generating functions, Normal distribution - Properties, Normal distribution - Standardizing normal random variables, Normal distribution - Moments, Normal distribution - Generating normal random variables, Normal distribution - The central limit theorem, Normal distribution - Infinite divisibility, Normal distribution - Stability, Normal distribution - Standard deviation, Normal distribution - Normality tests, Normal distribution - Related distributions, Normal distribution - Estimation of parameters, Normal distribution - Maximum likelihood estimation of parameters, Normal distribution - Unbiased estimation of parameters, Normal distribution - Occurrence, Normal distribution - Photon counting, Normal distribution - Measurement errors, Normal distribution - Physical characteristics of biological specimens, Normal distribution - Financial variables, Normal distribution - Lifetime, Normal distribution - Test scores

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Some of the properties of the normal distribution: If and a and b are real numbers, then (see expected value and variance). If and are independent normal random variables, then: Their sum is normally distributed with (proof). Their difference is normally distributed with . Both U and V are independent of each other. If and a ...

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Normal distribution, Normal distribution - Overview, Normal distribution - History, Normal distribution - Specification of the normal distribution, Normal distribution - Probability density function, Normal distribution - Cumulative distribution function, Normal distribution - Generating functions, Normal distribution - Properties, Normal distribution - Standardizing normal random variables, Normal distribution - Moments, Normal distribution - Generating normal random variables, Normal distribution - The central limit theorem, Normal distribution - Infinite divisibility, Normal distribution - Stability, Normal distribution - Standard deviation, Normal distribution - Normality tests, Normal distribution - Related distributions, Normal distribution - Estimation of parameters, Normal distribution - Maximum likelihood estimation of parameters, Normal distribution - Unbiased estimation of parameters, Normal distribution - Occurrence, Normal distribution - Photon counting, Normal distribution - Measurement errors, Normal distribution - Physical characteristics of biological specimens, Normal distribution - Financial variables, Normal distribution - Lifetime, Normal distribution - Test scores

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Normal distribution - Maximum likelihood estimation of parameters. Suppose are independent and identically distributed, and are normally distributed with expectation μ and variance σ2. In the language of statisticians, the observed values of these random variables make up a "sample from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on observed values of this sample. The joint probability de ...

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Normal distribution, Normal distribution - Overview, Normal distribution - History, Normal distribution - Specification of the normal distribution, Normal distribution - Probability density function, Normal distribution - Cumulative distribution function, Normal distribution - Generating functions, Normal distribution - Properties, Normal distribution - Standardizing normal random variables, Normal distribution - Moments, Normal distribution - Generating normal random variables, Normal distribution - The central limit theorem, Normal distribution - Infinite divisibility, Normal distribution - Stability, Normal distribution - Standard deviation, Normal distribution - Normality tests, Normal distribution - Related distributions, Normal distribution - Estimation of parameters, Normal distribution - Maximum likelihood estimation of parameters, Normal distribution - Unbiased estimation of parameters, Normal distribution - Occurrence, Normal distribution - Photon counting, Normal distribution - Measurement errors, Normal distribution - Physical characteristics of biological specimens, Normal distribution - Financial variables, Normal distribution - Lifetime, Normal distribution - Test scores

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Suppose you know that a given variable is exponentially distributed and you want to estimate the rate parameter λ. Exponential distribution - Maximum likelihood. The likelihood function for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from your variable, is where is the sample mean. The derivative of the likelihood function's logarithm is Consequently the maximum li ...

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Exponential distribution, Exponential distribution - Specification of the exponential distribution, Exponential distribution - Probability density function, Exponential distribution - Cumulative distribution function, Exponential distribution - Alternate specification, Exponential distribution - Occurrence and applications, Exponential distribution - Properties, Exponential distribution - Mean and standard deviation, Exponential distribution - Memorylessness, Exponential distribution - Quartiles, Exponential distribution - Entropy, Exponential distribution - Parameter estimation, Exponential distribution - Maximum likelihood, Exponential distribution - Bayesian inference, Exponential distribution - Generating exponential variates, Exponential distribution - Related distributions

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Exponential distribution - Mean and standard deviation. The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by In light of the examples given above, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call. The variance of X is . See also:

Exponential distribution, Exponential distribution - Specification of the exponential distribution, Exponential distribution - Probability density function, Exponential distribution - Cumulative distribution function, Exponential distribution - Alternate specification, Exponential distribution - Occurrence and applications, Exponential distribution - Properties, Exponential distribution - Mean and standard deviation, Exponential distribution - Memorylessness, Exponential distribution - Quartiles, Exponential distribution - Entropy, Exponential distribution - Parameter estimation, Exponential distribution - Maximum likelihood, Exponential distribution - Bayesian inference, Exponential distribution - Generating exponential variates, Exponential distribution - Related distributions

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The quality of an estimator is generally judged by its mean squared error. However, occasionally one chooses the unbiased estimator with the lowest variance. Efficient estimators are those that have the lowest possible variance among all unbiased estimators. In some cases, a biased estimator may have a uniformly smaller mean squared error than does any unbiased estimator, so one should not make too much of this concept. For that and other reasons, it is sometimes preferable not to limit oneself to unbiased estimators; see bias (statis ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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A consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows. An estimator tn (where n is the sample size) is a consistent estimator for parameter θ if and only if, for all ε > 0, no matter how small, we have It is called strongly consistent, if it c ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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Expected value - Linearity. The expected value operator (or expectation operator) E is linear in the sense that for any two random variables X and Y (which need to be defined on the same probability space) and any real numbers a and b. Expected value - Functional non-invariance. In general, the expectation operator and functions of random variables ...

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Expected value, Expected value - Mathematical definition, Expected value - Properties, Expected value - Linearity, Expected value - Functional non-invariance, Expected value - Non-multiplicativity, Expected value - Iterated expectation, Expected value - Inequality, Expected value - Representation, Expected value - Uses and applications of the expected value, Expected value - Expectation of matrices

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The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of X − E(X). The moments of some random variables can be used to specify their distributions, via their moment generating functions. To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and co ...

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Expected value, Expected value - Mathematical definition, Expected value - Properties, Expected value - Linearity, Expected value - Functional non-invariance, Expected value - Non-multiplicativity, Expected value - Iterated expectation, Expected value - Inequality, Expected value - Representation, Expected value - Uses and applications of the expected value, Expected value - Expectation of matrices

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Exponential distribution - Probability density function. The probability density function (pdf) of an exponential distribution has the form where λ > 0 is a parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0,∞). If a random variable X has this distribution, we write X ~ Exponential(λ). The exponential distributions can alternatively be parameterized by a scale parameter μ = 1/λ. ...

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Exponential distribution, Exponential distribution - Specification of the exponential distribution, Exponential distribution - Probability density function, Exponential distribution - Cumulative distribution function, Exponential distribution - Alternate specification, Exponential distribution - Occurrence and applications, Exponential distribution - Properties, Exponential distribution - Mean and standard deviation, Exponential distribution - Memorylessness, Exponential distribution - Quartiles, Exponential distribution - Entropy, Exponential distribution - Parameter estimation, Exponential distribution - Maximum likelihood, Exponential distribution - Bayesian inference, Exponential distribution - Generating exponential variates, Exponential distribution - Related distributions

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If X is an matrix, then the expected value of the matrix is a matrix of expected values: This property is utilized in covariance matrices. ...

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Expected value, Expected value - Mathematical definition, Expected value - Properties, Expected value - Linearity, Expected value - Functional non-invariance, Expected value - Non-multiplicativity, Expected value - Iterated expectation, Expected value - Inequality, Expected value - Representation, Expected value - Uses and applications of the expected value, Expected value - Expectation of matrices

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In Argentina, grades from 1 (sometimes 0) up to 10 are used, with some schools allowing decimals (up to 2) and some others only allowing whole numbers where: 10 (excellent) is the best possible grade 8-9.99 (very good) 6-7.99 (good) 4-5.99 (sufficient) up to 3.99 (insufficient) Most universities evaluate classes with two mid exams and a final. The final exam encompasses the whole course syllabus wheres the mid exams usually take just half. In some schools, if the average grade of ...

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Grade education, Grade education - Argentina, Grade education - Austria, Grade education - Belgium, Grade education - Central and Eastern Europe, Grade education - Chile, Grade education - Croatia, Grade education - Denmark, Grade education - Finland, Grade education - France, Grade education - Germany, Grade education - India, Grade education - International Baccalaureate, Grade education - Italy, Grade education - Iran, Grade education - The Netherlands, Grade education - Norway, Grade education - Peru, Grade education - Philippines, Grade education - Poland, Grade education - Portugal, Grade education - Russia, Grade education - Singapore, Grade education - Lower Primary Primary 1 to 3, Grade education - Upper Primary Primary 4 to 6, Grade education - Secondary Level for GCE O levels, Grade education - Junior College Level GCE A and AO levels, Grade education - Slovenia, Grade education - Sweden, Grade education - Switzerland, Grade education - Ukraine, Grade education - United States Canada England and Wales Hong Kong Australia New Zealand, Grade education - Yugoslavia former, Grade education - Related topics, Grade education - Related Links

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To build a model, several statistical "ingredients" need to be known. These are needed to ensure the estimator has some mathematical tractability instead of being based on "gut feel." The first is a set of statistical samples taken from a random vector (RV) of size N which can be put into a vector and their M parameters need to be established with their probability density function (pdf) or probabil ...

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Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. The estimator takes the measured data as input and produces an estimate of the parameters. It is also preferable to derive an estimator that exhibits optimality. An optimal estimator would indicate that all available information in the measured data has been extracted, for if there was unused informat ...

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Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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The notation MA(q) refers to a moving average model of order q. where the θ1, ..., θq are the parameters of the model and the εt, εt-1,... are again, the error terms. A moving average model is essentially a finite impulse response filter with some additional interpretation placed on it. ...

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Autoregressive moving average model, Autoregressive moving average model - Autoregressive model, Autoregressive moving average model - Example: An AR1-Process, Autoregressive moving average model - Calculation of the AR parameters, Autoregressive moving average model - Moving average model, Autoregressive moving average model - Autoregressive moving average model, Autoregressive moving average model - Note about the error terms, Autoregressive moving average model - Specification in terms of lag operator, Autoregressive moving average model - Fitting models, Autoregressive moving average model - Generalizations, Autoregressive moving average model - Reference

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ASK is the simplest kind of modulation that can be used to send data through a channel. It has several bad points: it can be used only when the signal-to-noise ratio is very high, because most of the signal is transmitted at reduced power, so it would be hard to recover. it needs A/D converters working at a frequency that could be higher than necessary: for example, if the bandwidth between 100 and 101 MHz is used for the transmission, the spectrum of the signal will be only 1 MHz wide, but the A/D converter will need to work at 101*2 = 202 MHz. In QAM modulation, ...

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Amplitude-shift keying, Amplitude-shift keying - Encoding, Amplitude-shift keying - Probability of error, Amplitude-shift keying - Considerations

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A major goal in the field of robots is self-replicating machines. Since all robots (at least in modern times) have a fair number of the same features, a self-replicating robot (or possibly a hive of robots) would need to do the following: Obtain construction materials Manufacture new parts Provide a consistent power source Program the new members To date, this has not been done. On a nanotechnical scale, nanomachines might also be designed to reproduce under their own power. This, in turn, has given rise to the "gray goo" theory of Armaggedon, as featu ...

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Biological reproduction, Biological reproduction - Asexual reproduction, Biological reproduction - Sexual reproduction, Biological reproduction - Mitosis and Meiosis, Biological reproduction - Reproductive strategies, Biological reproduction - Asexual vs. sexual reproduction, Biological reproduction - The Red Queen hypothesis, Biological reproduction - Life without reproduction, Biological reproduction - Mechanical reproduction

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