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Nakagami Distribution
The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling spread \Omega>0. Characterization Its probability density function (pdf) is : f(x;\,m,\Omega) = \fracx^\exp\left(-\fracx^2\right), \forall x\geq 0. where (m\geq 1/2,\text\Omega>0) Its cumulative distribution function is : F(x;\,m,\Omega) = P\left(m, \fracx^2\right) where ''P'' is the regularized (lower) incomplete gamma function. Parametrization The parameters m and \Omega are : m = \frac , and : \Omega = \operatorname \left ^2 \right Parameter estimation An alternative way of fitting the distribution is to re-parametrize \Omega and ''m'' as ''σ'' = Ω/''m'' and ''m''. Given independent observations X_1=x_1,\ldots,X_n=x_n from the Nakagami distribution, the likelihood function is : L( \si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakagami Pdf
Nakagami (written: 中上) is a Japanese surname. Notable people with the surname include: *, Japanese therapist * (1946–1992), Japanese writer, critic and poet * (born 1992), Japanese motorcycle racer See also * Nakagami District, Okinawa * Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ..., a statistical distribution {{surname, Nakagami Japanese-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivariate Normal
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Normal Table
A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal and then use the standard normal table to find probabilities. Normal and standard normal distribution Normal distributions In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is t ... are symmetrical, bell-shaped distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Normal Distribution
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables ''X'' and ''Y'', the distribution of the random variable ''Z'' that is formed as the ratio ''Z'' = ''X''/''Y'' is a ''ratio distribution''. An example is the Cauchy distribution (also called the ''normal ratio distribution''), which comes about as the ratio of two normally distributed variables with zero mean. Two other distributions often used in test-statistics are also ratio distributions: the ''t''-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable, while the ''F''-distribution originates from the ratio of two independent chi-squared distributed random variables. More general ratio distributions have been considered in the literature. Often the ratio distributions are heavy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Normal Distribution
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the class of ratio distributions, in which the numerator random variable has a degenerate distribution. Relation to original distribution In general, given the probability distribution of a random variable ''X'' with strictly positive support, it is possible to find the distribution of the reciprocal, ''Y'' = 1 / ''X''. If the distribution of ''X'' is continuous with density function ''f''(''x'') and cumulative distribution function ''F''(''x''), then the cumulative distribution function, ''G''(''y''), of the reciprocal is found by noting that : G(y) = \Pr(Y \leq y) = \Pr\left(X \geq \frac\right) = 1-\Pr\left(X<\frac\right) = 1 - F\left( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normally Distributed And Uncorrelated Does Not Imply Independent
In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables are normally distributed. This article demonstrates that assumption of normal distributions does not have that consequence, although the multivariate normal distribution, including the bivariate normal distribution, does. To say that the pair (X,Y) of random variables has a bivariate normal distribution means that every linear combination aX+bY of X and Y for constant (i.e. not random) coefficients a and b (not both equal to zero) has a univariate normal distribution. In that case, if X and Y are uncorrelated then they are independent. However, it is possible for two random variables X and Y to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modified Half-normal Distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Properties Using the \sigma parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by : f_Y(y; \sigma) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac, the probability density function is given by : f_Y(y; \theta) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. The cumulative distribution function (CDF) is given by : F_Y(y; \sigma) = \int_0^y \frac\sqrt \, \exp \left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: #With a shape parameter k and a scale parameter \theta. #With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/ \theta , called a rate parameter. In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which E 'X''= ''kθ'' = ''α''/''β'' is fixed and greater than zero, and E n(''X'')= ''ψ''(''k'') + ln(''θ'') = ''ψ''(''α'') − ln(''β'') is fixed (''ψ'' is the digamma function). Definitions The parameterization with ''k'' and ''θ'' appears to be more common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Gamma Distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution, the Weibull distribution and the gamma distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution. Characteristics The generalized gamma distribution has two shape parameters, d > 0 and p > 0, and a scale parameter, a > 0. For non-negative ''x'' from a generalized gamma distribution, the probability density function is : f(x; a, d, p) = \frac, where \Gamma(\cdot) denotes the gamma function. The cumulative distribution function is : F(x; a, d, p) = \frac , where \gamma(\cdot) denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-normal Distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Properties Using the \sigma parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by : f_Y(y; \sigma) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac, the probability density function is given by : f_Y(y; \theta) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. The cumulative distribution function (CDF) is given by : F_Y(y; \sigma) = \int_0^y \frac\sqrt \, \exp \left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Normal
In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix \Gamma, and the ''relation'' matrix C. The standard complex normal is the univariate distribution with \mu = 0, \Gamma=1, and C=0. An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: \mu = 0 and C=0 . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature. Definitions Complex standard normal random variable The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable Z whose real and imaginary parts are independent normally distributed random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
