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Math Help - Calculation of expected value...

  1. #1
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    Calculation of expected value...

    Hello all..
    Here it follows..
    X1, X2 are uniformly distributed over [0, 12]
    Y1, Y2 are uniformly distributed over [0, 20]
    We can define a new random variable D expressing the distance between both robots by D = sqrt(
    (X2 - X1)2 + (Y2 - Y1)2 ).
    What is the expected value of X1, i.e. E(X1)? How can the expected value of D be determined, either by an
    equation or approximated on a computer perhaps?
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  2. #2
    MHF Contributor matheagle's Avatar
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    1 I'm trying to figure out what is a Robot.
    2 I guess those 2's are squares.
    3 E(X1)=6, the midpoint of a uniform distribution.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by mathcuk View Post
    Hello all..
    Here it follows..
    X1, X2 are uniformly distributed over [0, 12]
    Y1, Y2 are uniformly distributed over [0, 20]
    We can define a new random variable D expressing the distance between both robots by D = sqrt(
    (X2 - X1)2 + (Y2 - Y1)2 ).
    What is the expected value of X1, i.e. E(X1)? How can the expected value of D be determined, either by an
    equation or approximated on a computer perhaps?
    As X_1, X_2, Y_1, Y_2 appear to be independed this is a (rather nasty) standard multi-dimensional integral. But the easiest way to estimate this integral is by a Monte-Carlo integration. This gives E(D) ~= 8.4 with SE ~= 0.04 from 10000 replications.

    The Monte-Carlo will even give a histogram of the sampling distribution (note: typo in the vertical axis legend; it's 10000 replications not 100000):

    CB
    Attached Thumbnails Attached Thumbnails Calculation of expected value...-gash.png  
    Last edited by CaptainBlack; April 25th 2009 at 10:43 AM.
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