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Math Help - Theoretical distribution of Y(x) and Elipson

  1. #1
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    Theoretical distribution of Y(x) and Elipson

    I'm a bit confused with this question. So the equation is Y=aX+Ei where:
    a: can be positive integer
    X: 1-20.
    Ei: Uniform distribution of integers from [-9, 9].

    It goes on to ask, show and discuss the theoretical distribution of Y|X and Ei.

    Well, wouldn't the theoretical distribution of Ei be uniformed? I'm not sure how am I suppose to show it? I'm not sure what the theoretical distribution of Y|X would be. From what the professor said, it should be uniformed too, but that doesn't really make any sense. It would make sense if x was uniformly distributed too, but x is just integers from 1-20.

    Thanks for your help!
    Last edited by Linnus; October 20th 2012 at 05:51 PM.
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  2. #2
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    Re: Theoretical distribution of Y(x) and Elipson

    Hey Linnus.

    What is the distribution of X? Is it just uniform as well or some other distribution?
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  3. #3
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    Re: Theoretical distribution of Y(x) and Elipson

    Hi Chiro,

    X is just 1,2,3,4,5,...all the way to 20 (is there a name for this type of distribution?). So the relationship between Y is X is how the the quality of the product goes up with increasing cost (x being the cost). Thanks!
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    Re: Theoretical distribution of Y(x) and Elipson

    Is it a random variable or is it a deterministic variable that can take on those values? Also if it is random and they all have the same choice, the distribution is a discrete uniform (as opposed to a continuous uniform distribution).
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  5. #5
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    Re: Theoretical distribution of Y(x) and Elipson

    X is a deterministic variable. Which means X must be 1,2,3,4,5,6...so on all the way to 20. Its given to you and set. You can't change it. Its not determined by picking a number out of a certain distribution. This is why I'm confused - how would Y|X have a uniform distribution (or any distribution at all) if the the X variable is not determined from a certain distribution.

    But the professor often gets confused in the class due to his age...so he could be mistaken.

    Thanks for your help.
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  6. #6
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    Re: Theoretical distribution of Y(x) and Elipson

    So your random variables have the relation Y = aX + Ei.

    In probability we have P(Y|X) = P(X|Y)*P(Y)/P(X) (You can look up Bayesian probability for this but its derived by using P(X|Y) = P(X and Y)/P(Y) and P(Y|X) = P(X and Y)/P(X)).

    Now consider your random variable Y: if E_i is a discrete uniform in [-a,a] then Ei + C where is a discrete uniform in [-a+C,a+c] where all values are integers. So Y is a discrete uniform distribution.

    X is a deterministic variable so what-ever value this takes on it will always have a probability of 1 (deterministic is just a special case of non-deterministic) so P(X) = 1.

    So we know P(Y) and P(X) and now P(X|Y) is always 1 since X is deterministic and does not change at all since X does not change given a Y (since it is constant).

    So now we have finally P(Y|X)
    = P(X|Y)*P(Y)/P(X)
    = P(X)*P(Y)/P(X) [Since P(X|Y) = P(X) as X doesn't ever depend on a particular Y value]
    = P(Y)

    where Y has a discrete uniform distribution.
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  7. #7
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    Re: Theoretical distribution of Y(x) and Elipson

    Thanks for the help. I got it. This class only covers regression so we didn't cover probability but I should have thought of it.
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  8. #8
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    Re: Theoretical distribution of Y(x) and Elipson

    Wait, quick question, Y isn't really described by Ei+C where C is a constant. Y is more accurately described by Ei+X where X is a variable that changes. I'm not sure why Y would be considered a constant if it changes with X? Thanks!
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