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While the magnetic moments (the black arrows) are oriented the same for both cases of γ, the precession is in opposite directions. Spin and magnetic moment are in the same direction for γ > 0 (as for protons). Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above. The ratio is conventionally ...
The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.
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 physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field.The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an external torque-exerting gravitational field.
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.
Gamma distribution example To illustrate how the product of moments yields a much simpler result than finding the moments of the distribution of the product, let , be sampled from two Gamma distributions, (;,) = with parameters =, whose moments are
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied ...
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.