|
%I #12 Mar 06 2017 10:47:13
%S 1,2,4,5,6,7,9,10,11,12,13,14,16,17,18,19,20,21,22,23,25,26,27,28,29,
%T 30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,54,
%U 55,56,57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,81,82,83,84,85,86,87,88
%N Position where the discrete difference of the Poissonian probability distribution function with mean n takes its lowest value. In case of a tie, pick the smallest value.
%C Is this the same as A183299?
%C Defining the probability for integer value x in a Poisson distribution of integer mean = q>0 as P(q,x) = e^(-q)*(q^x)/x! then P(n, a(n)+1)- P(n, a(n)) <= P(n, k+1)- P(n, k), for every k > 0.
%C (Very qualitative) Plot of a Poisson Distribution with mean n. The vertical line above a(n) indicates the place where the distribution has its minimum (negative) discrete difference.
%C P
%C ^
%C | *
%C | * *
%C | * *
%C | * *
%C | * *
%C | * | *
%C | * | *
%C | * | *
%C | * | *
%C *-------------+----+----------------------> x
%C n a(n)
%e With the definition of P(q,x) given in Comments, for instance, if n = 8 then
%e P(8, a(8)+1)-P(8, a(8)) = P(8,11)-P(8,10) = -0.027071
%e If we now calculate the discrete difference in a(n)+1 we then obtain
%e P(8,a(8)+2)-P(8,a(8)+1) = P(8,12)-P(8,11) = -0.0240634
%e and in a(n)-1
%e P(8,a(8))-P(8,a(8)-1) = P(8,10)-P(8,9) = -0.0248154
%e Both previous values are larger than the minimum obtained at a(n).
%t Table[Position[a=Differences@PDF[PoissonDistribution[n],Range[2^10]],Min@a]//Min,{n,1,80}]
%Y Cf. A183299.
%K nonn
%O 1,2
%A _Andres Cicuttin_, Feb 17 2017
|
