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  2. Moment-generating function - Wikipedia

    en.wikipedia.org/wiki/Moment-generating_function

    In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.

  3. Cumulant - Wikipedia

    en.wikipedia.org/wiki/Cumulant

    So the cumulant generating function is the logarithm of the moment generating function = ⁡ (). The first cumulant is the expected value ; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance ); but the higher cumulants are neither moments nor central moments, but ...

  4. Weibull distribution - Wikipedia

    en.wikipedia.org/wiki/Weibull_distribution

    The moment generating function of the logarithm of a Weibull ... The moment estimate of the scale parameter can then be found using the first moment equation as

  5. Normal distribution - Wikipedia

    en.wikipedia.org/wiki/Normal_distribution

    The moment generating function of a real random variable ⁠ ⁠ is the expected value of , as a function of the real parameter ⁠ ⁠. For a normal distribution with density ⁠ f {\displaystyle f} ⁠ , mean ⁠ μ {\displaystyle \mu } ⁠ and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to

  6. Taylor expansions for the moments of functions of random ...

    en.wikipedia.org/wiki/Taylor_expansions_for_the...

    In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.

  7. Gamma distribution - Wikipedia

    en.wikipedia.org/wiki/Gamma_distribution

    Indeed, we know that if X is an exponential r.v. with rate λ, then cX is an exponential r.v. with rate λ/c; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,these notes, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale).

  8. Geometric distribution - Wikipedia

    en.wikipedia.org/wiki/Geometric_distribution

    The moment generating function of the geometric distribution when defined over and respectively is [7] [6]: 114 = () = (), < ⁡ The moments for the number of failures before the first success are given by

  9. Probability-generating function - Wikipedia

    en.wikipedia.org/.../Probability-generating_function

    Other generating functions of random variables include the moment-generating function, the characteristic function and the cumulant generating function. The probability generating function is also equivalent to the factorial moment generating function , which as E ⁡ [ z X ] {\displaystyle \operatorname {E} \left[z^{X}\right]} can also be ...