N n x n x 1 μ n n 1 μ n x the expression may be

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n ! ( n x )! n x 1 μ n ¥ n 1 μ n ¥ x . The expression may be disassembled for the purpose of taking limits in the component parts: lim( n → ∞ ) n ! ( n x )! n x = lim( n → ∞ ) ( n ( n 1) · · · ( n x + 1) n x ) = 1 , lim( n → ∞ ) 1 μ n ¥ n = e μ , lim( n → ∞ ) 1 μ n ¥ x = 1 . On reassembling the parts, it is found that the binomial function has a limiting form of lim( n → ∞ ) b ( x ; n, p ) = μ x e μ x ! . This is the Poisson function. 10
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Poisson arrivals. Consider, for example, the impact of alpha particles upon a Geiger counter. Let f ( x, t ) denote the probability of x impacts in the time interval (0 , t ]. The following conditions are imposed: (a) The probability of a single impact in a very short time interval ( t, t + t ] is f (1 , t ) = a t , (b) The probability of more than one impact during that time interval is negligible, (c) The probability of an impact during the interval is independent of the impacts in previous periods. Consider the event of x impacts in the interval (0 , t + t ]. There are two possibilities: (i) All of the impacts occur in the interval (0 , t ] and none in the interval ( t, t + t ]. (ii) There are t 1 impacts in the interval (0 , t ] followed by one impact in the interval ( t, t + t ]. The probability of x impacts in the interval is therefore f ( x, t + t ) = f ( x, t ) f (0 , t ) + f ( x 1 , t ) f (1 , t ) = f ( x, t )(1 a t ) + f ( x 1 , t ) a t. Assumption (b) excludes all other possibilities. Assumption (c) implies that the probabilities of the mutually exclusive events of (i) and (ii) are obtained by multiplying the probabilities of the constituent events of the two sub-intervals. 11
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By rearranging the equation f ( x, t + t ) = f ( x, t )(1 a t ) + f ( x 1 , t ) a t, we get f ( x, t + t ) f ( x, t ) t = a © f ( x 1 , t ) f ( x, t ) .
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  • Spring '12
  • D.S.G.Pollock
  • Normal Distribution, Probability theory, probability density function, moment generating function

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