This question uses equation references from the textbook Bishop - Pattern Recognition And Machine Learning. I posted the 3 equation references from the book below the question.

2) (6 points) (Data marginal distribution for Gaussian mixture model) Consider a Gaussian mixture model in which the marginal distribution p(z) for the latent variable z is given by Bishop equation (9.10), and the conditional distribution p(xz) for the observed variable r is given by Bishop equation (9.11). Show that the marginal distribution p(x), obtained by summing p(2)p(xz) over all possible values of z, is a Gaussian mixture of the form given in Bishop equation (9.7). 2) (6 points) (Data marginal distribution for Gaussian mixture model) Consider a Gaussian mixture model in which the marginal distribution p(z) for the latent variable z is given by Bishop equation (9.10), and the conditional distribution p(xz) for the observed variable r is given by Bishop equation (9.11). Show that the marginal distribution p(x), obtained by summing p(2)p(xz) over all possible values of z, is a Gaussian mixture of the form given in Bishop equation (9.7)