Properties of Mathematical expectation

Properties of Mathematical expectation

(i) E(a) = a , where ‘a’ is a constant

(ii) E(aX) = aE(X)

(iii) E (aX + b) = aE(X ) + b , where ‘a’ and ‘b’ are constants.

(iv) If X ≥ 0,then E(X)≥ 0

(v) V (a) = 0

(vi) If X is random variable, then V (aX + b) = a ²V(X)

Concept of moments

The moments (or raw moments) of a random variable or of a distribution are the expectations of the powers of the random variable which has the given distribution.

Definition 6.11

If X is a random variable, then the r^th moment of X , usually denoted by μ_r , is defined as

provided the expectation exists.

Definition 6.12

If  X is a random variable, the r^th central moment of  X about a is defined as E[(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 = E[(X–μ_X)^r]

Note

·        μ'₁ = E(X) = μ_X , the mean of X.

·        μ₁ = E[X– μ_X] = 0.

·        μ₂ = E[(X– μ_X)²], the variance of X .

·        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.