By S. M. Ulam, A. R. Bednarek, Françoise Ulam
Many of the principles awarded keep their significance at the present time, and . . . are completely fundmental, either from a historic and from a systematic viewpoint.--Gian-Carlo Rota, Massachusetts Institute of expertise
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Additional info for Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators
Because the intervals l and Δl are nonoverlapping, the number of events that occur within one interval is independent of the number of events that occur within the other interval. Thus, the PDF of Lr can be obtained as follows: fLr ðlÞΔl 5 P½l , Lr # l 1 Δl 5 P½fXðlÞ 5 r 2 1g - fXðΔlÞ 5 1g 5 P½XðlÞ 5 r 2 1P½XðΔlÞ 5 1 8 9 < ðλlÞr21 = λr lr21 2λl e2λl fλ Δlg 5 e Δl 5 :ðr 2 1Þ! ; ðr 2 1Þ! Thus, fLr ðlÞ is given by fLr ðlÞ 5 λr lr21 2λl e ðr 2 1Þ! l $ 0; r 5 1; 2; . . which is the Erlang-r (or rth-order Erlang) distribution.
Which is the Erlang-r (or rth-order Erlang) distribution. The special case of r 5 1 is the exponential distribution. That is, fL1 ðlÞ 5 λ e2λl l$0 This result provides another definition of a Poisson process: It is a counting process with stationary and independent increments in which the intervals between consecutive events are exponentially distributed. 1 with the added constraint that the times between arrivals are exponentially distributed. 2 Definition of event intervals. Δl 40 Markov Processes for Stochastic Modeling Compound Poisson Process Let fNðtÞ; t $ 0g be a Poisson process with arrival rate λ.
We know that E½E½XjY 5 E½X. Also, because fXn ; n $ 0g is a martingale, E½Xn jℑ0 5 X0 . 1 Let X1 ; X2 ; . . be independent random variables with mean 0, and let Yn 5 We show that the process fYn ; n $ 1g is a martingale as follows: Pn k51 Xk . E½Yn11 jY1 ; Y2 ; . ; Yn 5 E½Yn 1 Xn11 jY1 ; Y2 ; . ; Yn 5 E½Yn jY1 ; Y2 ; . ; Yn 1 E½Xn11 jY1 ; Y2 ; . 2 Consider the variance of a sum of random variables. Specifically, let X0 5 0 and X1 ; X2 ; . . be independent and identically distributed random variables with mean E½Xk 5 0 and finite variance E½Xk2 5 σ2 ; k $ 1.
Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators by S. M. Ulam, A. R. Bednarek, Françoise Ulam