MLE mean and variance for Gaussian Univariate I'm trying to learn up some matlab to do some basic computer exercises and I have a small doubt in a question, the problem first arises in my very little knowledge in the subject of pattern recognition owing to my school schedule. I hope I don't come out as  a total noob, but just help me out with parts B and C from the below image

For the 2D asked in part b i just added x1+x2 for one data set, x2+x3 for the second and x1+x3 for the third. Is that the right way of doing it ? If not please tell me how the different pairing datasets should be formed.
Thanks
 A: When you say "added", do you mean you found the actual sums, as follows?
$$
\begin{array}{r}
0.42 - 0.087 \\
-0.2-3.3 \\
1.3-0.32 \\
0.39 + 0.71 \\
\vdots\quad\qquad{}
\end{array}
$$
If so, that doesn't get you the estimates for the two-dimensional Gaussian distribution.  What you need is the estimated mean and variance, and the variance is a $2\times 2$ matrix, sometimes called the "covariance matrix" because its entries are covariances (in particular, its diagonal entries are variances).  Notice the pair $(\mu,\Sigma)$.  The expected value $\mu$ is normally (no pun intended) thought of as a $2\times1$ column vector.  Its entries will be just the average for the first column in your table and the average for the second column.  The matrix
$$
\Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\  \sigma_{12} & \sigma_{22} \end{bmatrix}
$$
has as entries the variance $\sigma_{11}$ of the first scalar-valued random variable, the variance $\sigma_{22}$ of the second one, and the covariance $\sigma_{12}$ between them.
The MLE for $\sigma_{11}$ is
$$
\frac{1}{10}\sum_{i=1}^{10} (x_{1i} - \bar{x}_{1\bullet})^2
$$
where $$\bar{x}_{1\bullet}= \frac{1}{10}\sum_{i=1}^{10} x_{1i}$$
is the sample average for the first column.  (This differs from the conventional unbiased estimate in that the denominator is $10$ rather than $10-1$)  The MLE for $\sigma_{22}$ is found similarly by using the second column.  The MLE for $\sigma_{12}$ is
$$
\frac{1}{10}\sum_{i=1}^{10} (x_{1i}-\bar{x}_{1\bullet})(x_{2i}-\bar{x}_{2\bullet}).
$$
See this section of a Wikipedia article.
The article Estimation_of_covariance_matrices.
