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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, often used as the cumulative failure function ().. To accurately model failures over time, a cumulative failure distribution, () must be defined, which can be any cumulative distribution function (CDF) that gradually increases from to .
The Weibull distribution interpolates between the exponential distribution with intensity / when = and a Rayleigh distribution of mode = / when =. The Weibull distribution (usually sufficient in reliability engineering ) is a special case of the three parameter exponentiated Weibull distribution where the additional exponent equals 1.
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. [1] The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. [2] There are two equivalent parameterizations in common use:
In this example, a curve representing the exponential distribution overlays the distribution of AC failure times; the exponential distribution approximates the distribution of AC failure times. This particular exponential curve is specified by the parameter lambda, λ: λ = 1/(mean time between failures) = 1/59.6 = 0.0168.
A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution.
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.}