Moments

Moments 

If X is a random variable, the r^th moment of X, usually denoted by μ'_r is defined as
                                       μ'_r = δ[X']
if the expectation exists.
Note that μ'₁ = δ[X] = μ'_x the mean of X.

Central moments 

If X is a random variable, the r^th central moment of X about a is defined as δ[(X - a)^r] If a =  μ_x, we have the r^th  central moment of X about  μ'_x, denoted by  μ_r which is
                               μ_r = δ[(X - μ'_x)^r]

Note that  μ₁ = δ[(X - μ'_x)] = 0 and μ₂ = δ[(X - μ'_x)²], the variance of X. Also, note that all odd moments of X about μ'_x are 0 if the density function of X is symmetrical about μ'_x, provided such moments exist. In the ensuing few paragraphs we will comment on how the first four moments of a random variable or density are used as measures of various M .characteristics of the corresponding density. For some of these characteristics,  other measures can be defined in terms of quantiles.

Quantile 

The q^th quantile of a random variable X or of its corresponding distribution is denoted by ξ and is defined as the smallest number e satisfying F_x(ξ) ≥ q.
     If X is a continuous random variable, then the q^th quantile of X is given as the smallest number ξ satisfying F_x(ξ) = q. 

Median 

The median of a random variable X, denoted by med x , med (X), or ξ₅₀, is the .5th quantile.

Remark

In some texts the median of X is alternatively defined as any number, say med (X), satisfying P[X ≤ med (X)] ≥ 1/2 and P[X ≥ med (X)] ≥ 1/2· If X is a continuous random variable, then the median of X satisfies

                   ∫_{lim(-∞ ,med x)}f_x(x)dx = 1/2 =  ∫_{lim(med x , -∞)}f_x(x)dx