During one exposure, X-ray photons leave the tube

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Radiation physics and diagnostic imaging

Life at Thermal Energy, Ch Images from attenuation and from decay

Problem

During one exposure,

N_0 = 2.0\times10^{14}

X-ray photons leave the tube. Each photon independently has probability

p = 5.0\times10^{-11}

of being absorbed in one particular detector pixel, and is otherwise absorbed elsewhere or scattered away. Model the count

N

registered by the pixel as the sum of

N_0

independent yes/no contributions

X_i

, where

X_i = 1

if photon

i

is absorbed in the pixel and

X_i = 0

otherwise. ((a)) Write the mean and variance of one

X_i

in terms of

p

. ((b)) Using the fact that the variance of a sum of independent contributions is the sum of their variances (Chapter 2), show that

\mathrm{Var}(N) = \bar N(1-p)

, where

\bar N = N_0 p

is the mean count. ((c)) Evaluate

\bar N

and

\mathrm{Var}(N)

for the numbers given, and state to how many significant figures the variance equals the mean.

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