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As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random ...
The moment-generating function is given by: (; ... Handbook of mathematical functions with formulas, graphs, and mathematical tables (Unabridged and unaltered republ ...
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 {\textstyle f} , mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing. For k = 1, the density function tends to 1/λ as x approaches zero from above and is
This can be seen using moment generating functions as follows: () = = = = by the independence of the random variables. It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise.
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 ...
The moment generating function is given by = + ... Hence, the above formula can be used to estimate the noise variance in an MRI image from background data. [7] [8]
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).
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