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:
$\frac{1}{4\theta} \mathbb{1}(x \in [0, 2\theta]) + \frac{1}{4\theta} \mathbb{1}(x \in [0, 3\theta])$
for some unknown $\theta > 0$. For this problem, we call Unif$[0, 2\theta)$, the first component.
1. Compute the proportion $\pi$ of the first component.
2. Compute $E[X]$ and $V[X]$.
3. Assume that we start the $k$-th E-step of the EM algorithm with a candidate $\theta_k$ 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 $\theta_k$.
4. Assume that $n = 8$ and the observations are (in order)
1.01 1.02 1.19 1.19 1.28 2.39 2.56 2.58
and that the the EM algorithm is initialized at $\theta_0 = 3$, what are the values of the iterates: $\theta_1, \theta_2$ and $\theta_3$?
