Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

whereMXi(t)istheMGFoffi(x). Now

MXi

(

t

n

)

=1+

t

n

E[Xi]+^12

t^2

n^2

E[Xi^2 ]+···

=1+μi

t

n

+^12 (σ^2 i+μ^2 i)

t^2

n^2

+···,

and asnbecomes large

MXi

(

t

n

)

≈exp

(

μit

n

+^12 σ^2 i

t^2

n^2

)

,

as may be verified by expanding the exponential up to terms including (t/n)^2.

Therefore

MZ(t)≈

∏n

i=1

exp

(

μit

n

+^12 σ^2 i

t^2

n^2

)

=exp

(∑

iμi

n

t+^12

∑

iσ

2

i

n^2

t^2

)

.

Comparing this with the form of the MGF for a Gaussian distribution, (30.114),

we can see that the probability density functiong(z)ofZtends to a Gaussian dis-

tribution with mean

∑

iμi/nand variance

∑

iσ

2

i/n

(^2). In particular, if we consider
Zto be the mean ofnindependentmeasurements of thesamerandom variableX
(so thatXi=Xfori=1, 2 ,...,n) then, asn→∞,Zhas a Gaussian distribution
with meanμand varianceσ^2 /n.
We may use the central limit theorem to derive an analogous result to (iii)
above for theproductW=X 1 X 2 ···Xnof thenindependent random variables
Xi. Provided theXionly take values between zero and infinity, we may write
lnW=lnX 1 +lnX 2 +···+lnXn,
which is simply the sum ofnnew random variables lnXi. Thus, provided these
new variables each possess a formal mean and variance, the PDF of lnWwill
tend to a Gaussian in the limitn→∞, and so the productWwill be described
by a log-normal distribution (see subsection 30.9.2).
30.11 Joint distributions
As mentioned briefly in subsection 30.4.3, it is common in the physical sciences to
consider simultaneously two or more random variables that are not independent,
in general, and are thus described byjoint probability density functions. We will
return to the subject of the interdependence of random variables after first
presenting some of the general ways of characterising joint distributions. We
will concentrate mainly onbivariatedistributions, i.e. distributions of only two
random variables, though the results may be extended readily to multivariate
distributions. The subject of multivariate distributions is large and a detailed
study is beyond the scope of this book; the interested reader should therefore