Let's consider 2 dependent, normally distributed R.V.s, $X_1$ and $X_2$. The means, $\mu_1$ and $\mu_2$ are known, as well the covariance matrix $\Sigma$.
Let's consider the following random variable: $Y = w_1$$X_1 + w_2X_2$, where $w_1, w_2 \in [0,1]$ and $w_1 + w_2 = 1$. $w_1$ and $w_2$ are chosen in a way that the standard deviation of $Y$ is minimal.
Is there a best practice to build a confidence interval around $E[Y]$? We've been able to do it with bootstrap using matlab's: botci function, but we were wondering if there's an analytic solution.
A real world application of this scenario is the minimum variance portfolio of 2 risky assets, where $X_1$ and $X_2$ are the time series of the returns in a CER model.
EDIT: As pointed out in the comments, $Y$ is a gaussian RV, which makes things straightforward, and also the question pointless since the mean can be computed analytically: $E[Y] = w_1\mu_1 + w_2\mu_2$
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As pointed out in the comments, Y is a gaussian RV, which makes things straightforward, and also the question pointless since the mean can be computed analytically: $E[Y]=w_1*μ\mu1+w_2\mu_2$