Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS

− 6 − 4 − 2 234 6 8 10 12

0. 1

0. 2

0. 3

0. 4

σ=1

σ=2

σ=3

μ=3

Figure 30.13 The Gaussian or normal distribution for meanμ=3and

various values of the standard deviationσ.

− 4 − 2 0 −^2 −^1 ay^2

1

a 2 4

z z

φ(z)

Φ(z)

Φ(a)

Φ(a)

0. 2

0. 4

0. 6

0. 8

0. 1

0. 2

0. 3

0. 4

Figure 30.14 On the left, the standard Gaussian distributionφ(z); the shaded

area gives Pr(Z<a)=Φ(a). On the right, the cumulative probability function

Φ(z) for a standard Gaussian distributionφ(z).

distribution as

F(x)=Pr(X<x)=

1

σ

√

2 π

∫x

−∞

exp

[

−

1

2

(u−μ

σ

) 2 ]

du,

(30.107)

whereuis a (dummy) integration variable. Unfortunately, this (indefinite) integral

cannot be evaluated analytically. It is therefore standard practice to tabulate val-

ues of the cumulative probability function for the standard Gaussian distribution

(see figure 30.14), i.e.

Φ(z)=Pr(Z<z)=

1

√

2 π

∫z

−∞

exp

(

−

u^2

2

)

du. (30.108)