Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

Distribution Probability lawf(x)MGF E[X] V[X]

Gaussian

1

σ

√

2 π

exp

[

−

(x−μ)^2

2 σ^2

]

exp(μt+^12 σ^2 t^2 ) μσ^2

exponential λe−λx

(

λ

λ−t

)

1

λ

1

λ^2

gamma

λ

Γ(r)

(λx)r−^1 e−λx

(

λ

λ−t

)r

r

λ

r

λ^2

chi-squared

1

2 n/^2 Γ(n/2)

x(n/2)−^1 e−x/^2

(

1

1 − 2 t

)n/ 2

n 2 n

uniform

1

b−a

ebt−eat

(b−a)t

a+b

2

(b−a)^2

12

Table 30.2 Some important continuous probability distributions.

The probability density function for a Gaussian distribution of a random

variableX, with meanE[X]=μand varianceV[X]=σ^2 ,takestheform

f(x)=

1

σ

√

2 π

exp

[

−

1

2

(x−μ

σ

) 2 ]

. (30.105)

The factor 1/

√

2 πarises from the normalisation of the distribution,

∫∞

−∞

f(x)dx=1;

the evaluation of this integral is discussed in subsection 6.4.2. The Gaussian

distribution is symmetric about the pointx=μand has the characteristic ‘bell’

shape shown in figure 30.13. The width of the curve is described by the standard

deviationσ:ifσis large then the curve is broad, and ifσis small then the curve

is narrow (see the figure). Atx=μ±σ,f(x) falls toe−^1 /^2 ≈ 0 .61 of its peak

value; these points are points of inflection, whered^2 f/dx^2 = 0. When a random

variableXfollows a Gaussian distribution with meanμand varianceσ^2 ,wewrite

X∼N(μ, σ^2 ).

The effects of changingμandσare only to shift the curve along thex-axis or

to broaden or narrow it, respectively. Thus all Gaussians are equivalent in that

a change of origin and scale can reduce them to a standard form. We therefore

consider the random variableZ=(X−μ)/σ, for which the PDF takes the form

φ(z)=

1

√

2 π

exp

(

−

z^2

2

)

, (30.106)

which is called thestandard Gaussian distributionand has meanμ= 0 and

varianceσ^2 = 1. The random variableZis called thestandard variable.

From (30.105) we can define the cumulative probability function for a Gaussian