1000 Solved Problems in Modern Physics

10 1 Mathematical Physics

Poisson distribution

The probability thatxevents occur in unit time when the mean rate of occurrence
ism, is given by the Poisson distributionP(x).

P(x)=

e−mmx

x!

(x= 0 , 1 , 2 ,...) (1.64)

The distributionP(x) is normalized, that is

∑∞

x= 0

p(x)= 1 (1.65)

This is also a discrete distribution.
WhenNPis held fixed, the binomial distribution tends to Poisson distribution
asNis increased to infinity.
The expectation value, i.e.

〈x〉=m (1.66)

The S.D.,

σ=

√

m (1.67)

Properties:

pm− 1 =pm (1.68)

px− 1 =

x

m

pmandpx+ 1 =

m

m+ 1

px (1.69)

Normal (Gaussian distribution)

Whenpis held fixed, the binomial distribution tends to a Normal distribution asN
is increased to infinity. It is a continuous distribution and has the form

f(x)dx=

1

√

2 πσ

e−(x−m)

(^2) / 2 σ 2
dx (1.70)
wheremis the mean andσis the S.D.
The probability of the occurrence of a single random event in the intervalm−σ
andm+σis 0.6826 and that betweenm− 2 σandm+ 2 σis 0.973.
Interval distribution
If the data containsNtime intervals then the number of time intervalsnbetweent 1
andt 2 is
n=N(e−at^1 −e−at^2 ) (1.71)
whereais the average number of intervals per unit time. Short intervals are more
favored than long intervals.