Light as a Probability Wave — Question 9

Question 9

Fixed photon number produces binomial rather than Poisson partition noise

Exactly $N=400$ independent photons enter an interferometer. Each reaches detector D with probability $p=.250$. Detection is ideal, and K is the count at D. Use the binomial model.

1. Find the expected count and its standard deviation.

2. Find the standard deviation of the estimated fraction $\widehat p=K/N$ and the covariance with the complementary detector count.

Question 9 - Solution

Fixed photon number produces binomial rather than Poisson partition noise

(a) Method and calculation

$$\boxed{\langle K\rangle=Np=100},\quad\operatorname{Var}K=Np(1-p)=75.$$$$\boxed{\sigma_K=\sqrt{75}=8.66025}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_{\widehat p}=\sqrt{p(1-p)/N}=.0216506}.$$$$L=N-K\quad\Rightarrow\quad\boxed{\operatorname{Cov}(K,L)=-\operatorname{Var}K=-75}.$$

Physical check

The two output counts are anticorrelated because their sum is fixed. Poisson input fluctuations would define a different statistical experiment.