The attack of the academics...
Here is a rather academic problem which nevertheless reduces to some other interesting problems:
What is the expected value of the maximum of N random variables, each of which is between 0 and 1?
What is the expected maximum value of 2 variables under the constraint x+y=1?
If you have figured each of these out, you have passed the qualifying round for this month's IBM ponder this!
What is the expected value of the maximum of N random variables, each of which is between 0 and 1?
What is the expected maximum value of 2 variables under the constraint x+y=1?
If you have figured each of these out, you have passed the qualifying round for this month's IBM ponder this!
posted by littlecow at 1/04/2006 05:02:00 PM
12 Comments:
Bala, in the first question, do you mean that each of the N random variables are uniform between 0 and 1?
Yup! Assume a uniform distribution in q.2. also...
@grenade: right, ho! both the answers are right!
can u folks explain the physical meaning of 'expected value' ?
according to the n/n+1 solution, if there are 2 random variables, the expected maximum is 2/3 ? this is difficult to comprehend. is there a way to 'see' this visually / physically ?
expected value is just another name for average value.
think of it as a shooting game on a square dart board of side 1 unit. we are finding the centroid of the lower triangle formed by joining (0,0) to (1,1). why? because for all points on that triangle x>y...
...one more way of looking at it:
if the average is M, then equal number of cases exist where the max value occurs equidistant from M.
----x---y---M---- and -----y----x---M--- occur with a prob of M*M
---x----M---y---- occurs with a prob of M*(1-M)
---y----M---x---- occurs with a prob of M*(1-M)
Solve for M in M*M=2M*(1-M) to give M=2/3...
this is probably the boring rigorous way?
@grenade: you are right.
the second solution i gave is on shaky grounds and is incorrect. infact, if M!=1/2, it is not possible to find matching cases for certain values of x.
btw, have you seen the ancient tamil movie 'thiruvilayadal'? :-D
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@grenade,balakumar: i'm thinking the owvaiyaar conversation with muruga peruman on the fruit. probably a reference to the way u define boring rigorous. i may be mistaken.
i was reminded of the 'kalviyai nee ketkiraya naan ketkattuma?' conversation which sivaperuman has with the poet.
oh yeah .. the movie may itself be set in ancient times, but it is timeless, not to mention humorous.
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