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Expected Smaller (Posted on 2014-05-12) Difficulty: 3 of 5
Randomly select two numbers (not necessarily distinct) from the set

{0, 1/n, 2/n, ..., 1}

What is the expected value, in terms of n, of the smaller of the numbers?

Find the limit of this as n increases without bound.

No Solution Yet Submitted by Jer    
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Solution Continuous version Comment 2 of 2 |
a and b are uniformly distributed, independent random variables, 0 < a,b < 1
P:  Cumulative distribution function
p:  Probability density function
p is the derivative of P
P(a<x) = x
P(a>x) = 1-x
let m = min(a,b)
P(m>x) = P(a>x AND b>x) = (1-x)^2
P(m<x) = 1 - P(m>x) = 1 - (1 - 2x + x^2) = 2x - x^2
p(m) = 2 - 2x (by taking derivative)
note the integral of p(m) from 0 to 1 equals 1.

Call E(x) the expected value of x where x is the minimum of two numbers each of which is independent and uniformly distributed from 0 to 1.

E(x) is {integral of  x * p(m)} / {integral of  p(m)}  each integral is from 0 to 1.
E(x) = {integral of (2x - 2x^2) dx} / {integral of (2 - 2x) dx}
E(x) = {x^2 - 2(x^3)/3} / {2x - x^2}  each integral is from 0 to 1
E(x) = {1/3 - 0} / {1 - 0} = 1/3

Edited on May 14, 2014, 10:02 am
  Posted by Larry on 2014-05-14 10:00:57

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