and these quantities are called the kth central moments of X. Note the special cases
E[1] = 1, μ =E[X], σ^2 =E[(X−μ)^2 ] (11 .45)
The expectation of a sum of random variables X 1 , X 2 ,... , X nequals the sum of the
expectations and consequently
E(X 1 +X 2 +···Xn) = E(X 1 ) + E(X 2 ) + ··· +E(Xn) (11 .46)
The expectation of a product of independent random variables equals the product of
the expectations which is expressed
E(X 1 X 2 ···Xn) = E(X 1 )E(X 2 )···E(Xn) (11 .47)
Scaling
The probability density function f(x) is said to be symmetric with respect a
number x=μif for all values of xthe density function satisfies the relation
f(μ+x) = f(μ−x) (11 .48)
A random variable Xhaving a mean μand vari-
ance σ^2 can be scaled by introducing the new vari-
able Z= (X−μ)/σ. The variable Z is referred to
as the standardized variable corresponding to X.
Let f(x) denote the probability density function associated with the random
variable X and define the function f∗(z) = σf (x) = σf (σz +μ) as the probability
function associated with the random variable Z. Using the scaling illustrated in the
figure above, observe that x=σz +μwith dx =σ dz so that
f(x)dx =f(σz +μ)σ dz =f∗(z)dz
then the mean value on the Z-scale is given by
μ∗=
∫∞
−∞
zf ∗(z)dz =
∫∞
−∞
(x
σ
−μ
σ
)
f(x)dx
=
1
σ
∫∞
−∞
xf (x)dx −
μ
σ
∫∞
−∞
f(x)dx
=
1
σμ−
μ
σ(1) = 0
