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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?

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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:

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

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  • $\begingroup$ This assumes that each $X_{i,j}$ takes a single value (1 obervation), right? What if we have $m$ observations? $\endgroup$
    – user274779
    May 15, 2021 at 10:34
  • $\begingroup$ 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). $\endgroup$
    – Ben
    May 15, 2021 at 10:45
  • $\begingroup$ 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? $\endgroup$
    – user274779
    May 15, 2021 at 11:03
  • $\begingroup$ 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 $X_i \in \mathbb{R}^p$. $\endgroup$
    – Ben
    May 15, 2021 at 11:07
  • $\begingroup$ 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) $\endgroup$
    – user274779
    May 15, 2021 at 11:11

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