Log-concavity of probability of occurrence of at least r independent events

Statistics & Probability North-Holland

Letters 11 (1991) 63-64

January

1991

Log-concavity of probability of occurrence of at least r independent events Y.S. Sathe and S.M. Bendre Department

of Statistics,

University

of Bombay,

Vidyanagari,

Bomba?: 400 098, India

Received June 1989 Revised December 1989

Abstract:

For a sequence of n independent events, it is shown that the sequences and the probability of occurrence of at most r - 1 events are both log-concave. functions of order statistics follows as a corollary. Further, a counter example dependent events. Keywords: Log-concavity,

independent

of the probability of occurrence of at least r events The log-concavity of the sequence of distribution is given to show that the result does not hold for

events, order statistics.

1. Introduction

where U,_ = 1 and U,., = 0 for r > n, and

Let A,, . . . , A, be n independent events. Define U,,, as the probability of occurrence of at least r out of n events and V,,, as the probability of occurrence of at most r - 1 out of n events. It is proved that the sequences { q,n}z=o and { V;,n}z=O are both log-concave. The log-concavity of the sequence of distribution functions of r th order statistic from n independent random variables (Bapat and Beg, 1987) follows as a particular case. We further give a counter example to show that the result does not hold for dependent events in general.

v,,, = 1 - u,.,. Theorem.

For

p$-y

are ‘,e log-concave

> 9 >

;I

every fixed

n > 2, { U,,, }E 0 and functions

of r for

r =

.,

ur’n 2 Ur--l.t&l.n’ Kfn 2 Vr-l.nK+l.n holdforr=l,

2,....

Proof. For n = 2, U,z*=b*+P*-PlP2)2

2. Result

=P:4:+Pp224:+PIPZ(l-P,P*)+PlP2

A,, . . . , A, are n independent P(A,)=p,,

4,=1-p,,

events such that i=l,...,

n.

Let B,,, = at least rout

of n A,‘s occur,

0 1991 - Elsevier Science Publishers

= U2.2.

Thus the theorem is true for n = 2. Suppose it is true for n - 1. Since u,., =PnUrLI.n-, forral,

u,,, = P(B,,,), 0167-7152/91/$03.50

aPIP

B.V. (North-Holland)

n>2,

+ %U,,,-1

(1) 63

Volume

11, Number

therefore,

1

STATISTICS

& PROBABILITY

K-r+l.n

=Pn2Ur,n-1Ur-Z,n-l

+ Pn%Iw2,.&Jr+1,.-1 + Ur-l,n--lUr,n-l) ~P,2G,,-,

Kr.nWn-r+Z,n,

A by A

Vrfn > Vr-l,nVr+l,n. Hence

+43x2,-1

+ 2P,%7w1,.~Iu,,.-,) = u&.

2

which follows from (3) on interchanging and r by n-r+l. Thus, weget

+ C73J-l,“-1Ur+*.n-l

The inequality

1991

Then

for r > 2,

Ur-l,nUr+l,n

January

LETTERS

(2)

(2) follows from the fact that

U,f,& 2 Ur-l,n-IUr+l,n~l

the theorem.

0

Corollary. Let X,, X,, . _. , X, be n independent random variables, Ai = { X, < x } and further let F,.,(x) denote the c. d. f. of the rth smallest order statistics. Then F,,,(x)

= P@,,,).

and Hence by the above theorem, Ur:rJ-l

2 Ur-Z-n-1Ur.V1. F,fM

2 F,-&>F,+,,,(X>

For r = 1, from (l), and ur$ =p,‘+

2PJI,UI,,-1

>P,(P,

+ 4Jkn-1)

a

F,2,(4

4JJ2,,

u,.,.

The last inequality

follows

3 E-l.J4E+&>,

forr>l,

ltPJ4,4>

+ %kJkn-I = PJJ,,, +

+ 43-42,4

from the fact that

U,,, 2 U,,,, for n 2 2. Since the result is true for n = 2, by induction

whereF(.)=l-F(e).

0

The corollary is proved in Bapat and Beg (1987) using permanents. The theorem does not hold for dependent events in general, as can be seen from the following example. Example. Let P( A,) = 0.25, P( A2) = 0.25 P(A, f’ AZ) = 0.15. Then U,: -C U,,,.

u,‘, a Ur-l,nUr+l,n for all r and for all n > 2.

and

(3)

Now, consider V,., = P[at

64

most r - 1 out of n A,‘s occur]

=P[atleast

(n-r+l)

= Kr+l,n,

say.

A,‘soccur]

References Bapat, R.B. and M.L. Beg (1987), Order statistics for nonidentically distributed variables and permanents, Sankhy15 Ser. B, Submitted.