Begin2.DVI

Table 11.4 Mean and Variance for Discrete and Continuous Distributions

Discrete Continuous

population

μ=E[x] =

∑n

j=1

xjf(xj) mean μ=E[x] =

∫∞

−∞

xf (x)dx

μ

population

σ^2 =E[(x−μ)^2 ] =

∑n

j=1(xj−μ)

(^2) f(xj) variance σ (^2) =E[(x−μ) (^2) ] =
∫∞
−∞
(x−μ)^2 f(x)dx
σ^2

The continuous cumulative frequency function satisfies the properties

dF (x)

dx

=f(x), F (−∞ ) = 0, F (+ ∞) = 1 , and F(a)< F (b)if a < b (11 .43)

The table 11.4 illustrates the relationships of the mean and variance associated

with the discrete and continuous probability densities.

If X is a real random variable and g(X)is any continuous function of X, then

the numbers

E[g(X)] =

∑n

j=1

g(xj)f(xj) discrete

E[g(X)] =

∫∞

−∞

g(x)f(x)dx continuous

(11 .44)

associated with the probability density f(x)are defined as the mathematical expec-

tation of the function g(X). In the special case g(X) = Xkfor k= 1, 2 ,... , n an integer,

the equations (11.44) become

E[Xk] =

∑

j

xkjf(xj) discrete

E[Xk] =

∫∞

−∞

xkf(x)dx continuous

These expectation equations are referred to as the kth moment of X. In the special

case g(X) = (X−μ)k, the equations (11.44) become

E[(X−μ)k] =

∑

j

(xj−μ)kf(xj) discrete

E[(X−μ)k] =

∫∞

−∞

(x−μ)kf(x)dx continuous