Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS

expression for the PDF of thisZ does exist, but it is a rather complicated

combination of exponentials and a modified Bessel function.§

Two types of e-mail arrive independently and at random: external e-mails at a mean rate

of one every five minutes and internal e-mails at a rate of two every five minutes. Calculate

the probability of receiving two or more e-mails in any two-minute interval.

Let

X= number of external e-mails per two-minute interval,

Y= number of internal e-mails per two-minute interval.

Since we expect on average one external e-mail and two internal e-mails every five minutes
we haveX∼Po(0.4) andY∼Po(0.8). LettingZ=X+Ywe haveZ∼Po(0.4+0.8) =
Po(1.2). Now

Pr(Z≥2) = 1−Pr(Z<2) = 1−Pr(Z=0)−Pr(Z=1)

and

Pr(Z=0)=e−^1.^2 =0. 301 ,

Pr(Z=1)=e−^1.^2

1. 2

1

=0. 361.

Hence Pr(Z≥2) = 1− 0. 301 − 0 .361 = 0.338.

The above result can be extended, of course, to any number of Poisson processes,

so that ifXi=Po(λi),i=1, 2 ,...,nthen the random variableZ=X 1 +X 2 +

···+Xnis distributed asZ∼Po(λ 1 +λ 2 +···+λn).

30.9 Important continuous distributions

Having discussed the most commonly encountered discrete probability distri-

butions, we now consider some of the more important continuous probability

distributions. These are summarised for convenience in table 30.2; we refer the

reader to the relevant subsection below for an explanation of the symbols used.

30.9.1 The Gaussian distribution

By far the most important continuous probability distribution is theGaussian

ornormaldistribution. The reason for its importance is that a great many

random variables of interest, in all areas of the physical sciences and beyond, are

described either exactly or approximately by a Gaussian distribution. Moreover,

the Gaussian distribution can be used to approximate other, more complicated,

probability distributions.

§For a derivation see, for example, M. P. Hobson and A. N. Lasenby,Monthly Notices of the Royal

Astronomical Society, 298 , 905 (1998).