Question 3: Consider clustering 1D data with a mixture of 2 Gaussians…

Consider clustering 1D data with a mixture of 2 Gaussians using the EM algorithm. You are given the data points $x_1 = 1, x_2 = 10, x_3 = 20$.

Suppose the output of the E-step is as follows:

$\begin{gathered} \lambda_1^1 = 1, \; \lambda_2^1 = 0 \\ \lambda_1^2 = 0.4, \; \lambda_2^2 = 0.6 \\ \lambda_1^3 = 0.1, \; \lambda_2^3 = 0.9, \end{gathered}$

where $\lambda_C^i$ denotes the probability of the $i$-th data point to be in cluster $C$. After performing the M-step for the means $\mu_1$ and $\mu_2$, what are the updated means?