$($Gaussian Mixture Model $($GMM$))$ This question is about $($a simplified version of$)$ the Gaussian Mix$-$ ture Model $($GMM$),$ which is a popular model in statistics, data science and machine learning. For example, it is used in image processing and various clustering algorithms. Suppose that K is a discrete random variable that can either be $0$ or $1$ with probability $\backslash$pi $0$ and $\backslash$pi $1$ respectively, that is$,$

$(0,$ with probability $\backslash$pi $0=$ P $($K $=0),$

$1,$ withprobability$\backslash$pi $1=$P$($K$=1)=1\backslash$pi $0.$

K$=$

Conditional on K $=$ k with k in $\{0,1\},$ the distribution of X is N$(\backslash$mu k$,\backslash$sigma k$2),$ a normal distribution with

mean $\backslash$mu k and variance $\backslash$sigma k$2.$ That is$,$

$($a$)$ Derive the joint density of $($X$,$K$).$ State clearly the support of $($X$,$K$)$ in the joint density. Hint:

consider conditional distribution and the law of total probability.

$($b$)$ Denote the distribution of $($X$,$ K$)$ GMM$(\backslash$pi $0,\backslash$mu $0,\backslash$sigma $02,\backslash$mu $1,\backslash$sigma $12).$ Note that $\backslash$pi $1$ can be omitted as a parameter since $\backslash$pi $1=1\backslash$pi $0.$ Suppose that we have an i$.$i$.$d$.$ random sample of size n of these n pairs $($X$1,$ K$1),($X$2,$ K$2),...,($Xn$,$ Kn$).$ Each Xi belongs to either group $0$ or group $1$ depending on Ki$.$ Using part $($a$),$ derive the maximum likelihood estimator for all the five parameters $\backslash$pi $0,\backslash$mu $0,\backslash$sigma $02,\backslash$mu $1,\backslash$sigma $12.$ Hint: Let n$0=$ Pni$=11\{$Ki$=0\}$ and n$1=$ Pni$=11\{$Ki$=1\}$ be the number of Xi that belongs to group $0$ and group $1$ respectively. You may find expressing the likelihood function in terms of n$0$ and n$1$ useful.

$($c$)$ On Canvas, there is a dataset called $$GMM$.$csv$,$ which is a comma separated file. This dataset contains n $=100,000$ rows and $2$ columns. The first column is the realizations of K$,$ and the second column is the realizations of X$.$ Using any computing language of your choice, e$.$g$.$ Python$/$R$/$Julia$/$MATLAB$/$Excel$,$ compute the maximum likelihood estimators derived in part $($b$)$ using this dataset. Attach your code at the end of your assignment.