$3.$ Gaussian Mixture Model $($GMM$)$: $(45$ points$)$

$1.$ Choose the number of components $(\backslash ($ k $\backslash ))$ for the GMM algorithm based on the BIC score for each $\backslash ($ k $\backslash ).[$Hint$-$ Try a range of $\backslash ($ k $\backslash )$ values from $\backslash ($ k$=2\backslash )$ to $10](15)$

$2.$ Plot the BIC scores for different k values. Print the best value of k$.(5)$

$3.$ Implement the GMM algorithm to cluster the data into the selected number of components. $(5)$

$4.$ Print the final mean and covariance matrix for each component. $(5)$

$5.$ Visualize the data points and the GMM components as $2$D scatter plot. $(15)$