# 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 moment

**first order**is equal to 0.- μ1 = 0

The central moment

**second order**corresponds to the variance.- μ2 (μ) = E ((X − m1) 2)

The central moment

**third 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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