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Math Help - Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

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    Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

    Given t statistic

    $t = \frac{\bar{x} - \mu}{s/\sqrt(n)}$

    Could anyone mention the approach to prove $\bar(x)$ and $s$ are independent...you do not need to prove it...just tell the methods.

    As I knew that there are several methods to find $\bar(x)$ and $s$ are independent

    One approach that I have is to to simplify the process , let $n=2$ and under the assumption of normality.

    Then $s^2$ is proportional to $(x_1-x_2)^2$ and $\bar(x)$ to $x_1+x_2$. Now, $x_1-x_2$ and $x_1+x_2$ are jointly normal, so you only have to check that they are uncorrelated to show that they are independent. Finally, if $y,z$ are independent then so are $g(y),h(z)$ for any functions $g,h$.

    Another approach is that
    Let $X = {X_1, X_2, ..., X_n}$ ~ $N_n$, $X_i $are independent.
    Let $Y = {Y_1, Y_2, ..., Y_n}$ ~ $N_n$, $Y_i $are independent.

    $\bar{Y} = \sum_{i = 1}^n Y_i , s^2 = \sum_{i = 1}^n (Y_i - \bar{Y})^2/ (n-1)$
    The general approach is to show that $\bar{Y}$ can be written as a function of $X_1$ and $s^2$ can be written as a function of ${X_2, X_3,..., X_n}$
    Since $X_i$ are independent, $\bar{Y}$ and $s^2$are also independent.

    My question whether these two approaches the same. If not, do you think they work? Also, can you think of another approach?

    THANK YOU
    Last edited by askhwhelp; March 1st 2014 at 03:34 PM.
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  2. #2
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    Re: Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

    Hey askhwhelp.

    The easiest way to do this is if you have some function of two random variables (no matter what it is), if the probability P(X=x,Y=y) = P(X=x)P(Y=y) then you have independence. If the PDF is separable, then you have independence.

    Also if you can argue in an algebraic or other way that P(X|Y) = P(X) and P(Y|X) = P(Y) then you get the same end result. In other words, if knowing Y doesn't tell you anything about X and vice-versa, then you have done the exact same thing.

    Another way of thinking about this is to show that for any function Y = f(X), no function exists to relate the two variables together.

    All statements are all equivalent.
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    Re: Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

    Quote Originally Posted by chiro View Post
    Hey askhwhelp. The easiest way to do this is if you have some function of two random variables (no matter what it is), if the probability P(X=x,Y=y) = P(X=x)P(Y=y) then you have independence. If the PDF is separable, then you have independence. Also if you can argue in an algebraic or other way that P(X|Y) = P(X) and P(Y|X) = P(Y) then you get the same end result. In other words, if knowing Y doesn't tell you anything about X and vice-versa, then you have done the exact same thing. Another way of thinking about this is to show that for any function Y = f(X), no function exists to relate the two variables together. All statements are all equivalent.
    You are proposing three method, right?
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  4. #4
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    Re: Looking for approach to prove $\bar{x}$ and $s$ are independent for t-statistic

    Basically if X's are I.I.D and Y's are I.I.D then everything is independent to each other by definition (but possibly not identically distributed).

    Using that should prove the result.
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