Begin2.DVI

Standardization

The normal probability density function, sometimes called the Gaussian distri-

bution , has the form

N(x;μ, σ^2 ) = f(x) =^1

σ

√

2 π

e−

(^12) (x−σμ)^2
(11 .54)

and the area under this curve between the values x =a and x =b, where a < b,

represents the probability that a random variable X lies between the values aand b.

This probability is represented

P(a < X < b ) =

∫b

a

f(x)dx =

1

σ

√

2 π

∫b

a

e−

(^12) (x−σμ)^2
dx (11 .55)

Note that this integral cannot be integrated in closed form and so numerical inte-

gration techniques are used to create tables for a normalized form or standard form

associated with the above integral. See for example the table 11.5. The distribution

function associated with the normal probability density function is given by

F(x) =

1

σ

√

2 π

∫x

−∞

e−

(^12) (ξ−σμ)^2
dξ (11 .56)

Introducing the standardized variable z=x−μ

σ

, with dz =dx

σ

, the distribution func-

tion, given by equation (11.56), with variable xis converted to a normalized form

with variable z. The normalized form and associated scaling is illustrated below,

Φ(z) = √^1

2 π

∫z

−∞

e−z^2 /^2 dz, (11 .57)

and equation (11.54) is replaced by the standard form for the probability density

function

φ(z) = √^1

2 π

e−z^2 /^2 (11 .58)

which is the integrand of the integral given by equation (11.57). In the representa-

tions (11.58) and (11.57) the variable zis called a random normal number.