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Thread: Distribution of X+Y of a bivariate normally distributed (X,Y)

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    Distribution of X+Y of a bivariate normally distributed (X,Y)




    I need to give the distribution of $X+Y$ and $X-Y$ knowing that $(X,Y)$ is bivariate normally distributed with marginal means 1, marginal variances 1 and correlation $p=2$.Is it right that the marginal mean are simply $E(X)=1$ and $E(Y)=1$ ?I don't see where I should start knowing that.

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    Re: Distribution of X+Y of a bivariate normally distributed (X,Y)


    First of all, when you are talking about random vector, its mean is also a vector of the same dimension. In the context it seems to indicate that both X and Y has a marginal mean of 1, or equivalently the mean vector \begin{bmatrix} 1 \\ 1 \end{bmatrix} and so you should be correct. The marginal means are just the corresponding components in the mean vector.

    One more thing is that the correlation is a bounded quantity in [-1, 1]. So the correlation cannot be 2. Moreover, since the marginal variances are 1 too, so the absolute value of covariance |Cov[X, Y]| = |\rho|\sigma_X\sigma_Y \leq 1, i.e. also bounded by 1

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