Distrubution question

thechainsaw · #1 10-22-2006, 06:48 PM

You have 2 real numbers randomly distibuted between 0 and 1. What is the expected maximum?

Can you generalize for n numbers, where n is up to 10?

PairTheBoard · #2 10-22-2006, 07:27 PM

[ QUOTE ]
You have 2 real numbers randomly distibuted between 0 and 1. What is the expected maximum?

Can you generalize for n numbers, where n is up to 10?

[/ QUOTE ]

Assuming n independent uniform distributions for the numbers, letting M denote their maximum, then for any x in [0,1] P(M<x)=x^n.

PairTheBoard

thechainsaw · #3 10-22-2006, 08:03 PM

Thanks for that. But I was looking for something closely related. If I have 2 uniform independent distributions on [0,1], what % is likely to be the expected max.

(Basically I want to see the expected highest hand from 1 to 169 over a period of N trials - would it be correct to say that since P(1/2^1/2) = 0.5 for 2 goes, that the expected maximum is the square root of 2?)

bigpooch · #4 10-22-2006, 08:44 PM

If you have n independent identically distributed random
variables, each of which have uniform distribution over
the interval (0,1] (or [0,1] or [0,1) ), the maximum of
these will have a beta distribution. The mean or expected
value of the maximum is n/(n+1), so in this case, the mean
is 2/3.

PairTheBoard · #5 10-22-2006, 09:18 PM

[ QUOTE ]
[ QUOTE ]
You have 2 real numbers randomly distibuted between 0 and 1. What is the expected maximum?

Can you generalize for n numbers, where n is up to 10?

[/ QUOTE ]

Assuming n independent uniform distributions for the numbers, letting M denote their maximum, then for any x in [0,1] P(M<x)=x^n.

PairTheBoard

[/ QUOTE ]

You can figure it out from this because P(M<x)=x^n is the cumulative distribution function for M. So the density function for M is the derivitive, nx^(n-1) and the expected value for M is the integral,

int( xnx^(n-1) )[0,1] =
int( nx^n )[0,1] =
( (n/(n+1))x^(n+1) )[0,1] =
n/(n+1)
or 2/3 for n=2

Note: P(M<x)=x^n because the probability that any one of the numbers is less than x is x so the probability that all of them are less than x is x^n (independence). But all of them are less than x iff their maximum is less than x.

PairTheBoard