A282532 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.

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, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 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

Offset

1,2

Comments

Is this the same as A183299?

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.

(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.

P

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n a(n)

Links

Table of n, a(n) for n=1..80.

Example

With the definition of P(q,x) given in Comments, for instance, if n = 8 then
P(8, a(8)+1)-P(8, a(8)) = P(8,11)-P(8,10) = -0.027071
If we now calculate the discrete difference in a(n)+1 we then obtain
P(8,a(8)+2)-P(8,a(8)+1) = P(8,12)-P(8,11) = -0.0240634
and in a(n)-1
P(8,a(8))-P(8,a(8)-1) = P(8,10)-P(8,9) = -0.0248154
Both previous values are larger than the minimum obtained at a(n).

Mathematica

Table[Position[a=Differences@PDF[PoissonDistribution[n], Range[2^10]], Min@a]//Min, {n, 1, 80}]

Crossrefs

Cf. A183299.

Sequence in context: A139283 A188013 A183299 * A039243 A265187 A039186

Adjacent sequences: A282529 A282530 A282531 * A282533 A282534 A282535

Keyword

nonn

Author

Andres Cicuttin, Feb 17 2017

Status

approved