Quantitative measure of the rotational inertia of a body. As a rotating body spins about an external or internal axis (either fixed or unfixed), it opposes any change in the body's speed of rotation that may be caused by a torque. It is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of its distance from the axis of rotation.
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If X has a continuous probability density function f(x) then the moment generating function is given by
where F is the cumulative distribution function.
If X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and
where the ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi and the moment-generating function for Sn is given by
where t is a vector and is the dot product.
Provided the moment-generating function exists in an open interval around t = 0, the nth moment is given by
If the moment generating function is finite in such an interval, then it uniquely determines a probability distribution.
The cumulant-generating function is the logarithm of the moment-generating function.