# A Maximum Likelihood Estimate of the mean for a multivariate normal distribution, the MLE of the mean is a scalar value or a vector scalar values?

In the following equation, I have found the maximum likelihood estimate of the mean for a multivariate normal distribution

$\therefore \mu^*_{MLE}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$

But I am wondering if the $\mu$ is a scalar value or a vector scalar values.

• Is this the right way to think of what it is? $\mu_{MLE} = \begin{bmatrix}\mu_1\\\mu_2\\\vdots\\\mu_n\end{bmatrix} = \begin{bmatrix}\dfrac{1}{n}\sum_{i=1}^n x_1\\\dfrac{1}{n}\sum_{i=1}^n x_2\\\vdots\\\dfrac{1}{n}\sum_{i=1}^n x_i\end{bmatrix} = \begin{bmatrix}E(X_1)\\E(X_2)\\\vdots\\E(X_n)\end{bmatrix} = \begin{bmatrix}\bar{X_1} \\ \bar{X_2} \\ \vdots \\ \bar{X_n}\end{bmatrix}$ – user122358 Mar 12 '17 at 13:43

## 2 Answers

Mean of multivariate normal distribution is a vector of length equal to the number of variables. It cannot be scalar. Say you would use multivariate normal distribution to describe human weight, height and the average number of hours slept - single mean of those three variables would be meaningless.

If $x_{i}$ is vector-valued, $\mu$ is a vector (one mean for each entry of $x_{i}$). You may consider the scalar case as a special case in which $\mathrm{dim}(x_{i})=1$.