A288347 -id:A288347 - cp's OEIS frontend

A288346 Median of 2^X + 2^Y where X and Y are independent random variables with B(n,1/2) distributions.

3, 4, 6, 9, 12, 20, 24, 40, 64, 80, 128, 160, 256, 320, 528, 768, 1088, 1536, 2176, 3072, 4352, 6144, 9216, 12288, 18432, 32768, 36864, 65536, 73728, 131072, 163840, 264192, 327680, 532480, 655360, 1064960, 1310720, 2162688, 2621440, 4325376, 6291456, 8650752

Offset: 1

Matt Frank, Jun 08 2017

nonn

Interpretation: Start with a portfolio of stocks A and B each worth $1, and flip a pair of coins. Stock A doubles if the first coin is heads and otherwise stays constant. Stock B doubles if the second coin is heads and otherwise stays constant. This sequence gives your median portfolio value after n pairs of coin flips.
Although a median of integers can be a half-integer, as an empirical observation only integers appear in this sequence.
The mean of 2^X + 2^Y is 2(3/2)^n.

Robert Israel, Table of n, a(n) for n = 1..5299

Cf. A288347, A288416, which are additive rather than multiplicative.

f:= proc(n)
local PX, i, pt, j;
for i from 0 to n do PX[i]:= binomial(n,i)/2^n od:
pt:= 0:
for j from 0 while pt^2 < 1/2 do pt:= pt + PX[j] od:
j:= j-1:
pt:= (pt-PX[j])^2:
for i from 0 do
  pt:= pt + 2*PX[j]*PX[i];
  if pt = 1/2 then error("Probability 1/2 for i=%1 j=%2",i,j) fi;
  if pt > 1/2 then return(2^i + 2^j) fi
od:
end proc:
map(f, [$1..60]); # Robert Israel, Jun 21 2017

TwoToThe[x_] := 2^x;
WeightsMatrix[n_] := Table[Binomial[n, i] Binomial[n, j], {i, 0, n}, {j, 0, n}]/2^(2 n);
ValuesMatrix[n_, f_] := Table[f[i] + f[j], {i, 0, n}, {j, 0, n}];
Distribution[n_, f_] := EmpiricalDistribution[Flatten[WeightsMatrix[n]] -> Flatten[ValuesMatrix[n, f]]];
NewMedian[n_, f_] := Mean[Quantile[Distribution[n, f], {1/2, 1/2 + 1/2^(2 n)}]];
Table[NewMedian[n, TwoToThe], {n, 42}]

A288416 Median of (2X-n)^2 + (2Y-n)^2 where X and Y are independent random variables with B(n, 1/2) distributions.

Original entry on oeis.org

2, 4, 2, 4, 10, 8, 10, 8, 10, 16, 18, 20, 18, 20, 26, 20, 26, 20, 26, 32, 26, 32, 34, 36, 34, 36, 34, 40, 34, 40, 50, 40, 50, 52, 50, 52, 50, 52, 50, 52, 58, 64, 58, 64, 58, 68, 58, 68, 74, 68, 74, 72, 74, 72, 82, 80, 82, 80, 82, 80, 82, 80, 90, 100, 90, 100

Offset: 1

Matt Frank, Jun 09 2017

nonn

Interpretation: Start at the origin, and flip a pair of coins. Move right one unit if the first coin is heads, and otherwise left one unit. Then move up one unit if the second coin is heads, and otherwise down one unit. This sequence gives your median squared-distance from the origin after n pairs of coin flips.
The mean of (2X-n)^2 + (2Y-n)^2 is 2n, or A005843.
A continuous analog draws each move from N(0,1) rather than from {+1,-1}, so the final x- and y- coordinates are distributed as N(0,sqrt(n)). Then the final point has probability 1 - exp(-r^2/2n) of being within r of the origin, and the median squared-distance for this continuous analog is n log(4). We also observe empirically that for this discrete sequence, a(n)/n approaches log(4).

			For n=3 the probabilities of ending up at the lattice points in [-3,3]x[-3,3] are 1/64 of:
1 0 3 0 3 0 1
0 0 0 0 0 0 0
3 0 9 0 9 0 3
0 0 0 0 0 0 0
3 0 9 0 9 0 3
0 0 0 0 0 0 0
1 0 3 0 3 0 1
So the squared-distance is 2 with probability 36/64, 10 with probability 24/64, and 18 with probability 4/64; the median squared-distance is therefore 2.

Cf. A288347, which is similar, with shifted coordinates; and also A288346.

Shifted[x_, n_] := (2 x - n)^2;
WeightsMatrix[n_] := Table[Binomial[n, i] Binomial[n, j], {i, 0, n}, {j, 0, n}]/2^(2 n);
ValuesMatrix[n_, f_] := Table[f[i, n] + f[j, n], {i, 0, n}, {j, 0, n}];
Distribution[n_, f_] := EmpiricalDistribution[Flatten[WeightsMatrix[n]] -> Flatten[ValuesMatrix[n, f]]];
NewMedian[n_, f_] :=
Mean[Quantile[Distribution[n, f], {1/2, 1/2 + 1/2^(2 n)}]];
Table[NewMedian[n, Shifted], {n, 66}]

cp's OEIS Frontend

A288346 Median of 2^X + 2^Y where X and Y are independent random variables with B(n,1/2) distributions.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica