# Let $X_1,...,X_n \sim N_p(\mu,\Sigma)$ (iid), how do I interpret the sample mean $\bar{X}=\frac{1}{n}\sum^n_{i=1}X_i?$

Let $$X_1,...,X_n \sim N_p(\mu,\Sigma)$$ (iid). From this question I asked before, I understood that an $$X_i$$ is a gaussian vector, and each vector, once observed (say $$n$$ times), can be stored as a $$n\times p$$ matrix.

Now, how do I interpret the sample mean $$\bar{X}=\frac{1}{n}\sum^n_{i=1}X_i?$$ In particular, how do you sum up $$n$$, $$n\times p$$ matrices. Is my reasoning wrong somewhere?

Each $$X_i$$ is a vector with length $$p$$ and the sample mean will also be a vector of length $$p$$. In particular, if we denote the elements of each vector as $$X_i = (X_{i,1},...,X_{i,p})$$ then the mean vector is:

$$\bar{X} = \Bigg( \frac{1}{n} \sum_{i=1}^n X_{i,1},...,\frac{1}{n} \sum_{i=1}^n X_{i,p} \Bigg).$$

• This assumes that each $X_{i,j}$ takes a single value (1 obervation), right? What if we have $m$ observations?
– user274779
May 15, 2021 at 10:34
• Your question stipulates $n$ observations from a $p$-dimensional multivariate normal distribution. If that is correct then each $X_i$ is a vector of length $p$ (i.e., it contains $p$ scalar random variables).
– Ben
May 15, 2021 at 10:45
• I'm a bit confused. Can't the $X_i$'s be $m\times p$ matrices, given we have $m$ obervations for each of them? So $X_1,...,X_n$ would mean $n$, $m\times p$ matrices, given we observed all $X_i$'s $m$ times? Or this is not correct, as in when we write $X_1,...,X_n \sim N_p(\mu,\Sigma)$, this means that each $X_i \in \mathbb{R}^p$ is just $1$ obervation?
– user274779
May 15, 2021 at 11:03
• The notation $N_p$ usually refers to the multivariate normal distribution producing random vectors of length $p$ (and nothing in your question mentions a value $m$). It is certainly possible to produce random matrices instead, but this is inconsistent with the notation used in your question. From what you have written in your question, you should have $X_i \in \mathbb{R}^p$.
– Ben
May 15, 2021 at 11:07
• yes, I have written that $X_i\in \mathbb{R}^p$. But I also asked what would be the situation if these $X_i$'s were matrices? (I didn't mention $m$, but I did mention $n$ as in $n\times p$ matrirces)
– user274779
May 15, 2021 at 11:11