On monoticity of some binomial probabilities

(1)

U.U.D.M. Report 2010:12

Department of Mathematics

Uppsala University

On monoticity of some binomial probabilities

Sven Erick Alm

(2)

(3)

ON MONOTICITY OF SOME BINOMIAL PROBABILITIES

SVEN ERICK ALM

Abstract. We prove a monotonicity property of the binomial distribution.

1. Introduction

In [1], Ruci´nski and R¨odl, in their study of hypergraph perfect matchings, conjecture that

P (Xr ≤ r − 1) > P (X1= 0),

for all r = 2, . . . , k, where Xr∼ Bin(k − d, r/k), k ≥ 2 and 1 ≤ d < k.

By introducing

fr = P (Xr< r),

their conjecture is equivalent with the conjecture that the sequence {fr} attains a unique

minimum at r = 1.

We show that this is indeed the case, but prove the stronger result that the sequence is strictly increasing in r for r ≤ k − d.

2. The result

The conjecture follows from the following lemma and the trivial observation that fr ≡ 1 for r > k − d.

Lemma 2.1. The sequence {fr, 1 ≤ r ≤ k} is strictly increasing in r for r ≤ k − d.

Proof. The proof is based on the following natural coupling. Let U1, . . . , Uk−d be i.i.d.

uniform on (0, 1) and construct Xr= k−d X i=1 I(Ui< r k), r = 1 . . . , k, where I is the indicator function. Further, let

Ym= (Xr | Xr+1 = m) ∼ Bin m, r r + 1 . For r ≥ 1, fr = P (Xr< r) = P (Xr+1< r) + k−d X m=r P (Xr+1= m) · P (Ym< r). Date: June 22, 2010. 1

(4)

2 SVEN ERICK ALM

Note that, for m ≥ r, P (Xr+1= m) ≤ P (Xr+1= r), with strict inequality for m > r. If

r = k − d, we get fr= P (Xr+1 < r) + P (Xr+1 = r) · P (Yr< r) < P (Xr+1 ≤ r) = fr+1, and if r < k − d, fr< P (Xr+1 < r) + P (Xr+1= r) · k−d X m=r P (Ym < r),

so that the lemma is proved if we can show that the last sum is at most 1.

Write Ym =Pmi=1Vi, where V1, . . . , Vm are i.i.d. Bernoulli with parameter r/(r + 1),

and introduce Nr= min{n : Yn= r} ∼ NegBin r, r r + 1 , with E(Nr) = r · r+1_r = r + 1. Then,

k−d X m=r P (Ym< r) ≤ ∞ X m=r P (Ym < r) = ∞ X m=r P (Nr> m) = E(Nr) − r−1 X 0 P (Nr> m) = r + 1 − r = 1,

which proves the lemma. References

[1] Andrzej Ruci´nski & Vojtˇech R¨odl, When are hypergraph perfect matchings as easy as fractional perfect matchings? In preparation (2010).

Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06, Uppsala, Sweden.