Problem $10$ Let $x_1,$dots,$x_n$ be independent copies of the random variable $x$ where $x$

is a mixture of two uniform random variables and has pdf:

$f(x)=\frac{1}{4}I(xin[0,2])+\frac{1}{4}1(xin[,3])$

for some unknown $>0.$ For this problem, we call Unif $[0,2),$ the first component.

Compute the proportion $$ of the first component.

Compute $E[x]$ and $V[x].$

Assume that we start the $k-$th E$-$step of the EM algorithm with a candidate ${}_k$

from the previous $M$ step. Let $w_1,$dots,$w_n$ be the weights obtained in the E$-$step

of the EM algorithm. Compute these weights and show that they can take only

three values depending on $x_i$ and ${}_k.$

Assume that $n=8$ and the observations are $($in order$)$

and that the the EM algorithm is initialized at ${}_0=3,$ what are the values of the

iterates: ${}_1,{}_2$ and ${}_3?$