Can someone please help me solve this question? If possible I would like to have clear steps with the solution so that I can have a better understanding.

A univariate Gaussian Mixture Model (GMM) has two components, both of which have their own mean and standard deviation. The model is defined by the following parameters: if z = 0 Z ~ Bernoulli(a) = 1-a ifz =1 Xn | Z = 0 ~ N(3v, 4w?) Xn | Z = 1 ~ N(2v, 9w?) For a dataset of N datapoints, find the following: 2.1.1. Write the marginal probability of x, i.e. p(I) [3pts] - Express your answers in terms of M(x|3v, 4w?) and M(x|2v, 9w?) may be simpler HINT: For this question suppose we have a Gaussian Distribution M(3v, 7w?), it means N(u = 3v, q2 = 7w2) - HINT: Start with the Sum Rule 2.1.2. E-Step: Compute the posterior probabilities, i.e, p(zo|x), p(z1|x) [3pts] - Express your answers in terms of M(3v, 4w2) and N(2v, 9w?) - HINT: Try to apply Beyes Rule 2.1.3. M-Step: Compute the updated value of w. (You can keep / fixed when you calculate the derivative.) [9pts] -- Note that w is a shared variable between the two distributions, your final answer should be one equation including both Gaussian distributions Express your answers in terms of T, x, and v (you will need to expand )(3v, 4w?) and N(2v, 9w2) into its PDF form) - HINT: Start from the below equation, note that O is shorthand for various variables, and take the derivative w.r.t. w2 N Z ((0 z) = EEp(zk(In, Gota) In [p(In, Zx|0)] N Z l(v, w), a| z) = _Ep(zk | In, Gota) In [p(In, Zk | HK, oh, a) ] N =EEP(ZA | En, Gold) In [p( zx | a)p(In | zk, V, w/)] Recall that p(In | Zk, V, w?) -+ N(In | /k, The) has been defined at the beginning of the