Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS

Now using Φ(−z)=1−Φ(z)gives

Φ

(

μ− 140

σ

)

=1− 0 .030 = 0. 970.

Using table 30.3 again, we find

μ− 140

σ

=1. 88. (30.113)

Solving the simultaneous equations (30.112) and (30.113) givesμ= 173.5,σ=17.8.

The moment generating function for the Gaussian distribution

Using the definition of the MGF (30.85),

MX(t)=E

[

etX

]

=

∫∞

−∞

1

σ

√

2 π

exp

[

tx−

(x−μ)^2

2 σ^2

]

dx

=cexp

(

μt+^12 σ^2 t^2

)

,

where the final equality is established by completing the square in the argument

of the exponential and writing

c=

∫∞

−∞

1

σ

√

2 π

exp

{

−

[x−(μ+σ^2 t)]^2

2 σ^2

}

dx.

However, the final integral is simply the normalisation integral for the Gaussian

distribution, and soc= 1 and the MGF is given by

MX(t)=exp

(

μt+^12 σ^2 t^2

)

. (30.114)

We showed in subsection 30.7.2 that this MGF leads toE[X]=μandV[X]=σ^2 ,

as required.

Gaussian approximation to the binomial distribution

We may consider the Gaussian distribution as the limit of the binomial distribu-

tion when the number of trialsn→∞but the probability of a successpremains

finite, so thatnp→∞also. (This contrasts with the Poisson distribution, which

corresponds to the limitn→∞andp→0 withnp=λremaining finite.) In

other words, a Gaussian distribution results when an experiment with a finite

probability of success is repeated a large number of times. We now show how

this Gaussian limit arises.

The binomial probability function gives the probability ofxsuccesses inntrials

as

f(x)=

n!

x!(n−x)!

px(1−p)n−x.

Taking the limit asn→∞(andx→∞) we may approximate the factorials by

Stirling’s approximation

n!∼

√

2 πn

(n

e

)n