16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 7 Last time: Moments of the Poisson distribution from its generating function G(S) =e u(s-1) ds d-G r= aG u +A =+H-p Example: Using telescope to measure intensity of an object Photon flux photoelectron flux. The number of photoelectrons are poisson distributed. During an observation we cause N photoelectron emissions. Nis the measure of the signal s=N=at u=it 1(S For signal-to-noise ratio of 10, require N=100 photoelectrons. All this follows from the property that the variance is equal to the mean. This is an unbounded experiment, whereas the binomial distribution is for n number of trials 9/30/2004955AM Page 1 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 1 of 10 Lecture 7 Last time: Moments of the Poisson distribution from its generating function. ( 1) ( 1) 2 2 ( 1) 2 1 2 2 2 1 1 2 2 22 2 2 ( ) s s s s s s Gs e dG e ds d G e ds dG X ds d G dG X ds ds X X X µ µ µ µ µ µ µ µ σ µ µµ µ − − − = = = = = = = = = + = + = − = +− = = Example: Using telescope to measure intensity of an object Photon flux Î photoelectron flux. The number of photoelectrons are Poisson distributed. During an observation we cause N photoelectron emissions. N is the measure of the signal. 2 2 1 N N N SN t t S t t t S t λ σ µλ λ λ σ λ λ σ = = = = = = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ For signal-to-noise ratio of 10, require N = 100 photoelectrons. All this follows from the property that the variance is equal to the mean. This is an unbounded experiment, whereas the binomial distribution is for n number of trials