and the variance on the Z-scale is given by

σ∗^2 =

∫∞

−∞

(z−μ∗)^2 f∗(z)dz =

∫∞

−∞

z^2 f∗(z)dz since μ∗= 0

σ∗^2 =

∫∞

−∞

(

x−μ

σ

) 2

f(x)dx =

1

σ^2

∫∞

−∞

(x−μ)^2 f(x)dx =

1

σ^2 σ

(^2) = 1

This demonstrates that the introduction of a scaled variable Z centers the mean at

zero and introduces a variance of unity.

The Normal Distribution

The normal probability distribution is a continuous function with two parameters

called μand σ > 0. The parameter σis called the standard deviation and σ^2 is called

the variance of the distribution. The normal probability distribution has the form

f(x) = N(x;μ, σ^2 ) =

1

σ

√

2 π

e−

(^12) (x−μ) (^2) /σ 2

1
σ
√
2 π
exp
[
−
1
2 (x−μ)
(^2) /σ 2
]
, −∞ < x < ∞
(11 .49)

and is illustrated in the figure 11-7. The parameter μis known as the mean of

the distribution and represents a location parameter for positioning the curve on

the x-axis. Note the normal probability curve is symmetric about the line x=μ.

The parameter σis sometimes called a scale parameter which is associated with the

spread and height of the probability curve. The quantity σ^2 represents the variance

of the distribution and σrepresents the standard deviation of the distribution. The

total area under this curve is 1 with approximately 68.27% of the area between the

lines μ±σ, 95.45% of the total area is between the lines μ± 2 σand 99.73% of the

total area is between the lines μ± 3 σ. The area bounded by the curve N(x;μ, σ^2 )and

the x-axis is unity. The area under the curve N(x;μ, σ^2 )between X=band X=a < b

represents the probability P(a < X ≤b). For example, one can write the probabilities

P(μ−σ < X ≤μ+σ) =. 6827

P(μ− 2 σ < X ≤μ+ 2σ) =. 9545

P(μ− 3 σ < X ≤μ+ 3σ) =. 9973

(11 .50)

The function

φ(z) = N(z; 0,1) = √^1

2 π

e−z^2 /^2 (11 .51)

is called a normalized probability distribution with mean μ= 0 and standard devia-

tion of σ= 1.