Usage of sample covariance and sample mean

Question

I understand the difference between sample mean/covariance and population mean/covariance and how to calculate them. However, I'm a bit unsure about what happens afterwards. If I only have the sample mean and sample covariance, do I use them the same way as the population mean/covariance for all other calcualations or do I have to take the fact that I have the sample and not the population in consideration?

For example, lets say I have $ X_1, X_2, X_3 $ which are independent and normally distributed. They have $ \underline{\bar{x}} = ( 1, 2, 3) $ and an unbiased sample covariance matrix $ S$ (with dimension 3x3). Now, if I want to calculate the distribution of $$ Y = X_1 -2X_2 + 3X_3 = \begin{pmatrix} 1 & -2 & 3\end{pmatrix} \begin{pmatrix} X_1\\ X_2 \\X_3 \end{pmatrix} = AX$$ can I use the regular formula $$\mu_y = A\underline{\mu} \\ \Sigma_y = A\Sigma A^T$$ and just switch $\underline{\mu} $ to $\underline{\bar{x}} $ and $\Sigma$ to $ S $ ?

My guess is that it works, but that I get the sample mean and sample covariance for $Y$, not the population mean/covariance. $$\bar{y} = A\underline{\bar{x}}\\S_y= ASA^T$$

If anyone can clear this out for me that would be very helpful.

Answer 1

You'll never know the true distribution of $Y$ because you don't know the true joint distribution of $X_1,X_2,X_3$. What you do is approximating the probability distributions you need, which is perfectly OK. However, we always need to keep in mind that the PDF we get afterwards is an estimation. In a broader sense, you assume (or know) that your samples are coming from normal distribution and do a ML or MAP estimate for the population parameters (together with bias correction), which are $\mu$ and $\Sigma$.