Generalize Proposition as follows. For [math]i=1,\dots,d[/math] let [math]X_i\sim\mathcal{N}(\mu_i,\sigma_i)[/math] be independent Gaussian random variables. Let [math]\lambda_i\not=0[/math] be real numbers. Show that [math]X:=\lambda_1X_1+\cdots+\lambda_dX_d[/math] is again a Gaussian random variable with mean [math]\mu=(\mu_1+\cdots+\mu_d)/d[/math] and [math]\sigma^2=\lambda_1^2\sigma_1^2+\cdots+\lambda_d^2\sigma_d^2[/math].