EM Algorithm for Mixtures of Lines [3 P]

Assume that the training examples ${\bf x}_n \in \mathbb{R}^2$ with $n=1,...,N$ were generated from a mixture of $K$ lines

$$P(x_{n,2} \vert z_{n,k}=1) = \mathcal{N}( x_{n,2} \vert \theta_{k,1} x_{n,1} + \theta_{k,2},\sigma_k)$$ (7)

where

$$\mathcal{N}( x \vert \mu,\sigma) = \frac{1}{\sqrt{2\pi} \sigma} \exp \left( -\frac{(x-\mu)^2}{2 \sigma^2}\right)$$ (8)

and the hidden variable $z_{n,k}=1$ if ${\bf x}_n$ is generated from line $k$ and 0 otherwise. Derive the update equations for the M-step of the EM algorithm for the variables ${\bf\theta}_{k}$ and $\sigma_k$ .