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lecture07 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 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 ) ) 2 2( 1 ) 2 1 2 2 2 1 1 2 222 22 () s s s s s s Gs e dG e ds dG e ds dG X ds X ds ds XX X µ µµ σ = = = = = = == =+ =− = = 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 St t t S t λ σµ λσ ⎛⎞ = ⎜⎟ ⎝⎠ For signal-to-noise ratio of 10, require 100 N = 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.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 2 of 10 3. The Poisson Approximation to the Binomial Distribution The binomial distribution, like the Poisson, is that of a random variable taking only positive integral values. Since it involves factorials, the binomial distribution is not very convenient for numerical application. We shall show under what conditions the Poisson expression serves as a good approximation to the binomial expression – and thus may be used for convenience. () ! ( 1 ) !! kn k n bk p p knk =− Consider a large number of trials, n , with small probability of success in each, p , such that the mean of the distribution, np , is of moderate magnitude. 1 2 1 2 1 2 1 2 1 2 Define with large and small Recalling: ! ~ 2 Stirling's formula lim 1 ! 1 !( )! 2 1 ! 2( ) ! 1 n n n n nk k k n n nk k n n np n p p n nn e e n n knk n n ne nk e n k n µ π µµ µπ + →∞ + + −+ + + = ⎛⎞ ⎜⎟ ⎝⎠ ≈− = 1 2 1 as becomes large relative to ! 1 ! k k kk k n k e n e kee e k = The relative error in this approximation is of order of magnitude 2 Rel. Error ~ k n
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 3 of 10 However, for values of k much smaller or larger than µ , the probability becomes small. The Normal Distribution Outline: 1. Describe the common use of the normal distribution 2. The practical employment of the Central Limit Therorem 3. Relation to tabulated functions Normal distribution function Normal error function Complementary error function 1. Describe the common use of the normal distribution Normally distributed variables appear repeatedly in physical situations.
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lecture07 - 16.322 Stochastic Estimation and Control, Fall...

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