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In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the Euler–Mascheroni constant), and the standard deviation is /
The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time [citation needed]. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.
The uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable. This method is very useful in theoretical work.
The Erlang distribution is the distribution of the sum of k independent and identically distributed random variables, each having an exponential distribution. The long-run rate at which events occur is the reciprocal of the expectation of X , {\displaystyle X,} that is, λ / k . {\displaystyle \lambda /k.}
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables.Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
If a random variable X has a probability density function then the characteristic function is its Fourier transform with sign reversal in the complex exponential [3] [page needed]. [4] This convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform. [5]