Let random variables $X$ and $Y$ be independent Normal with distributions $N(\mu_{1},\sigma_{1}^2)$ and $N(\mu_{2},\sigma_{2}^{2})$. Show that the distribution of $(X,X+Y)$ is bivariate Normal with mean vector $(\mu_{1},\mu_{1}+\mu_{2})$ and covariance matrix

$$ \left( \begin{array}{ccc} \sigma_{1}^2 & \sigma_{1}^2 \\ \sigma_{1}^2 &\sigma_{1}^2+\sigma_{2}^2 \\ \end{array} \right).$$

Thanks .

  • 2
    $\begingroup$ This question has several aspects: (1) show that $(X,X+Y)$ is Normal, (2) compute the means, and (3) compute the variance-covariance matrix. For which of these do you need guidance? $\endgroup$
    – whuber
    Nov 8, 2013 at 21:03

1 Answer 1

  1. Is $(X, X+Y)$ normal? Yes! It is a linear combination of independent univariate normal distributions.
  2. Means: the mean of $X$ is $\mu_1$, and the mean of $X+Y$ is the sum of the means because they are independent, so $\mu_1+\mu_2$.
  3. Variance-covariance. The variance of the sum of two independent random variables is the sum of their variance. So, the variance of $(X,X+Y)$ is $(\sigma_1^2, \sigma_1^2+\sigma_2^2)$. Now calculate the covariance:

$$ Cov(X,X+Y) = Cov(X,X)+Cov(X,Y) = \sigma_1^2+0$$

A multidimension normal distribution is defined by its mean and variance-covariance matrix. Therefore,

$$ (X,X+Y)\sim N \left(\left( \begin{array}{c} \mu_1 \\ \mu_1+\mu_2 \end{array}\right), \left( \begin{array}{ccc} \sigma_{1}^2 & \sigma_{1}^2 \\ \sigma_{1}^2 &\sigma_{1}^2+\sigma_{2}^2 \\ \end{array} \right) \right) $$


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