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 '13 at 21:03
  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,

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.