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  2. Exponential distribution - Wikipedia

    en.wikipedia.org/wiki/Exponential_distribution

    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 ...

  3. Relationships among probability distributions - Wikipedia

    en.wikipedia.org/wiki/Relationships_among...

    If X and Y are independent exponential random variables with mean μ, then X − Y is a double exponential random variable with mean 0 and scale μ. If X i are independent Bernoulli random variables then their parity (XOR) is a Bernoulli variable described by the piling-up lemma. (See also ratio distribution.)

  4. Exponential family - Wikipedia

    en.wikipedia.org/wiki/Exponential_family

    The terms "distribution" and "family" are often used loosely: Specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; [a] however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families ...

  5. Distribution of the product of two random variables - Wikipedia

    en.wikipedia.org/wiki/Distribution_of_the...

    The distribution of the product of a random variable having a uniform distribution on (0,1) with a random variable having a gamma distribution with shape parameter equal to 2, is an exponential distribution. [18]

  6. Cumulative distribution function - Wikipedia

    en.wikipedia.org/wiki/Cumulative_distribution...

    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 .

  7. Cumulant - Wikipedia

    en.wikipedia.org/wiki/Cumulant

    The natural exponential family of a distribution may be realized by shifting or translating K(t), and adjusting it vertically so that it always passes through the origin: if f is the pdf with cumulant generating function () = ⁡ (), and | is its natural exponential family, then () = (), and () = (+) ().

  8. Weibull distribution - Wikipedia

    en.wikipedia.org/wiki/Weibull_distribution

    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.

  9. Compound probability distribution - Wikipedia

    en.wikipedia.org/wiki/Compound_probability...

    Compounding a half-normal distribution with its scale parameter distributed according to a Rayleigh distribution yields an exponential distribution. This follows immediately from the Laplace distribution resulting as a normal scale mixture; see above. The roles of conditional and mixing distributions may also be exchanged here; consequently ...