What are moments in probability theory

Moments of distribution functions

Moments are parameters of a distribution function of a random variable. They correspond to the parameters of descriptive statistics. The terms expected value, variance, skewness and curvature for describing a function result from the so-called central moments (see there). A distribution function is determined by specifying all of its moments, if they exist. There are also distributions whose moments do not exist, such as B. the Lévy distribution. One distinguishes ordinary moments, absolute, central and the Moment at c.
Example: A normal distribution is determined, for example, by its expected value and its second moment, since all odd-numbered moments vanish and the higher even-numbered moments are directly related to the second moment.

Definition - Ordinary Moments

Let X be a random variable, k a natural and r a real number. Then it is referred to as ordinary moment of order k with respect to r (or simply as the kth ordinary moment) the expected value of the kth power of the random variable "centered" on r:
mk (r) = E ((X − r) k)

Continuous random variable

With a continuous real random variable with the probability density function fX we get:
mk (r): = - ∞∫∞ (x − r) kfX (x) dx

Discrete random variable

mk (r) = i = 1∑∞ (xi −r) kpi

ordinary moments (order k)

m1 = E (x)
m2 = E (x2) = Var (x) + (E (x)) 2

Absolute moments

Mk (r) = E (∣ (X − r) ∣k)
is referred to as k th absolute moment of x with respect to r.

Central moments

The central moments substitute the expected value E (X) itself for r.
μk: = E ((X − m1) k)
The central momentfirst order is equal to 0.
μ1 = 0
The central momentsecond order corresponds to the variance.
μ2 (μ) = E ((X − m1) 2)
The central momentthird order corresponds to the skew with γ ∗ σ3.

Moment and the characteristic function

By deriving the formula for the characteristic function several times, one obtains a representation of the ordinary moments through the characteristic function as:
E (Xk) = ikφX (k) (0) (k = 1,2, ...)

Moment around a constant (c)

  • the moment around c (c: constant, kth order): E (x − c) k

Moments around zero

If r = 0, one speaks of Moments around zero, or designated
mk = mk (0) = E ((X − 0) k) = E (Xk)
simply as the kth moment. The kth moment can be determined with the moment generating function.

Mathematics as a subject is so serious that no opportunity should be missed to make it more entertaining.

Blaise Pascal

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