Poisson Process Theorem

Theorem 1 Under the following assumptions, Yt number of occurrences in the time 

Yt counts the number of occurrences of the event in the time interval (0, t). 

Occurrences of an event happen randomly through time at a constant rate A. 

What happens in one time interval is independent of what happens in another non-overlapping time interval, (irulependent increments) 

The probability of exactly one event occurring in a very short time interval t, h) is approximately proportional to the length of the interval h. 

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It is not difficult to recognize the renewal process in the Poisson process discussed earlier, where the random variable Thad a y(l, v) exponential distribution, as a consequence of which N, (or, with the earlier notation X) turned out to have a /7(vf) Poisson distribution. The Poisson process represents one of the few cases where the distribution of the renewal process is known for any value of t. The importance of the central limit theorem for renewals lies in the very fact that, for large enough values of f, one can count on normal distribution even if the exact distribution of Nf is unknown. The validity of the theorem can be easily checked in this particular case, since by substitution one gets the well-known result that for large enough /i the approximation U(fi) N(fi, h) can be used. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

Ions are attracted to metal or other conductors that have no charge on them. The attraction arises from a process of induction. An ion in the neighborhood of a conducting surface induces a field. That field attracts the ion to the surface. Induction is readily modelled by recognizing an extraordinary mathematical property, the uniqueness of differential equations such as Poisson s equation. According to the uniqueness theorem, if you can find any solution to Poisson s equation that satisfies the boundary conditions for the problem of interest, you have found the only solution, even if you found it by wild guesses or clever tricks. 

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