From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Variance of max(X,Y)
Date: 30 Dec 1999 19:19:21 GMT
Newsgroups: sci.math

In article <386638FF.74170B13@vguard.com>,
Isak Levinson  <isak@vguard.com> wrote:

>I'd like to know the Variance of max(X,Y).

>what is the relation to Var(X), Var(Y).

>I suspect that it should be less than max(Var(X), Var(Y)).

No, it isn't.  For example, consider a probability space with three
outcomes: 1) with probability p has X = 1/p, Y = 0 2) with
probability p has X = 0, Y = 1/p 3) with probability 1-2p has X=Y=0
(where p is small and positive). Then Var(X) = Var(Y) = 1/p^2 - 1,
but Var(max(X,Y)) = 2/p^2 - 4. What you can say is that
Var(max(X,Y)) <= Var(X) + Var(Y), and this example shows that
estimate is essentially the best possible.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Variance of max(X,Y)
Date: 30 Dec 1999 21:55:56 GMT
Newsgroups: sci.math

In article <84gb7p$jlq$1@nntp.itservices.ubc.ca>,
Robert Israel <israel@math.ubc.ca> wrote:
> What you can say is that
>Var(max(X,Y)) <= Var(X) + Var(Y), and this example shows that
>estimate is essentially the best possible.

Actually there is an improvement possible:

Var(max(X,Y)) + Var(min(X,Y)) <= Var(X) + Var(Y)

Does this inequality have a name?

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2