Robert Mann - An Introduction to Particle Physics and the Standard Model · Problem 7

A Toy Theory — Robert Mann

Physics — Particle Physics · poisson-distribution-photomultiplier

Problem

In a photomultiplier tube, each incident photon produces

n

electrons. Over

N

repetitions of this process the average number of electrons produced is

\bar{n} = \frac{1}{N}\sum_{i=1}^{N} n_i

and the probability of observing a particular number

n

of electrons for a typical measurement is

P(n) = \frac{\bar{n}^n}{n!}e^{-\bar{n}}

a distribution called a Poisson distribution. (a) Show that

\sum_n P(n) = 1

. (b) Compute the average

\langle n \rangle = \sum_n n P(n)

\sqrt{\left\langle (n - \langle n \rangle)^2 \right\rangle}

of the number of electrons observed.