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Math Help - Question about problem statement (marginal distribution)

  1. #1
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    Question about problem statement (marginal distribution)

    I am doing some problems from a practice final and would like to know if the following problem has mistakes in the way it is written. We are supposed to apply a corollary that doesn't seem to have any relevance in this context. It is throwing me off.


    Problem statement: Suppose that $X$ is $N(\mu,\sigma^2)$ and $Y$ is $N(\mu,\sigma^2)$ and they are independent. Let $U=X+Y$ and $V=X+Y$. Use the following corollary to find the marginal distributions of $X$ and $Y$.


    Corollary: Let $X_1, \ldots, X_n$ be mutually independent random variables with $X_i$ is $n(\mu_i, \sigma_i^2)$. Let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be fixed constants Then


    $Z=\sum_{i=1}^n(a_iX_i + b_i)$ is $n(\sum_{i=1}^n(a_i\mu_i + b_i),\sum_{i=1}^na_i^2\sigma_i^2)$.


    Also, aren't the marginal distributions of $X$ and $Y$ just $X$ and $Y$ themselves, because they are independent of each other??


    Any help would be greatly appreciated. My final is tomorrow and I'm studying as hard as I can.
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  2. #2
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    Re: Question about problem statement (marginal distribution)

    Hey abscissa.

    Do you mean finding the marginal distributions of X and Y given a joint distribution U = f(X,Y)? As you have pointed out since X and Y are independent then the joint distribution is P(X = x, Y = y) = P(X=x)*P(Y=y) and integrating out one will give the other.

    Use the theorem to show that the sum of normals has the means summed and the variances summed as well.
    Thanks from abscissa
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