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Math Help - Expected value of balls in box (a challenging one)

  1. #1
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    Expected value of balls in box (a challenging one)

    If we have a box that contains 1000000 balls numbered from 1 to 1000000.
    if we take "K" balls without replacement. let X be the maximum number and y be the minimum number in the K balls.
    What is the expected value E(x-y) as function of "K".
    Last edited by billobillo; April 30th 2012 at 01:47 PM.
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    Re: Expected value of balls in box (a challenging one)

    Quote Originally Posted by billobillo View Post
    If we have a box that contains 1000000 balls numbered from 1 to 1000000.
    if we take "K" balls without replacement. let X be the maximum number and y be the minimum number in the K balls.
    What is the expected value E(x-y) as function of "K".
    @billobillo
    You will never understand this question unless you yourself sit and work with the numbers.
    May I suggest that you cut the question down to 100 balls an let K=8.
    List the smallest eight numbers. What is x=?~,~y=?~,~x-y=?

    List the larest eight numbers. What is x=?~,~y=?~,~x-y=?

    List any eight numbers. What is x=?~,~y=?~,~x-y=?

    Now what is the maximum for x?
    Now what is the minimum for y?
    Now what is ?\le x-y\le ??

    The think about number partitions?

    When you have done that. Tell us what you find.
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  3. #3
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    Re: Expected value of balls in box (a challenging one)

    Quote Originally Posted by Plato View Post
    @billobillo
    You will never understand this question unless you yourself sit and work with the numbers.
    May I suggest that you cut the question down to 100 balls an let K=8.
    List the smallest eight numbers. What is x=?~,~y=?~,~x-y=?

    List the larest eight numbers. What is x=?~,~y=?~,~x-y=?

    List any eight numbers. What is x=?~,~y=?~,~x-y=?

    Now what is the maximum for x?
    Now what is the minimum for y?
    Now what is ?\le x-y\le ??

    The think about number partitions?

    When you have done that. Tell us what you find.
    Hi Plato,

    Sorry I know this is not my problem, but I've been thinking on it lately. I don't quite understand where you're planning to lead our OP, so may I ask for a precision ?

    What do you mean by "what is the maximum for x" ? Why would it help ?

    Thanks
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    Re: Expected value of balls in box (a challenging one)

    The title is a bit misleading...

    Try it Moo, but maybe not start with K=8...
    if K=2, you draw 2 balls out of 100, what is the expected value of the bigger numbered ball?
    Now you draw 3 balls...
    Then you should get it by now.

    prove by using principle of symmetry
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    Re: Expected value of balls in box (a challenging one)

    Quote Originally Posted by Moo View Post
    Sorry I know this is not my problem, but I've been thinking on it lately. I don't quite understand where you're planning to lead our OP, so may I ask for a precision ?
    What do you mean by "what is the maximum for x" ? Why would it help ?
    The set \{1,2,3,4,5,6,7,8\} has x=8,~y=1,~\&~x-y=7

    The set \{93,94,95,96,97,98,99,100\} has x=100,~y=93,~\&~x-y=7.

    The set \{1,24,37,45,50,61,87,100\} has x=100,~y=1,~\&~x-y=99

    Clearly 7\le~x-y~\le 99

    Now \mathcal{P}(x-y=99)=\dfrac{\binom{98}{6}}{\binom{100}{8}}.
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    Re: Expected value of balls in box (a challenging one)

    Thanks for your replies, before posting this topic, I worked on a sample of 7 balls with K=3 and I was able to find E(x) and E(y) as function of K but not E(x-y),
    I'm not a math specialist but I guess that x and y are not independent events, any suggestions?
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    Re: Expected value of balls in box (a challenging one)

    E(x-y) is always E(x)-E(y), you are confusing with E(XY) which required independence to factorize
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