Question

A mixture of m univariate Gaussians has the PDF: X(x) - where each pi 0 and Σ-i pi-1, and N(x; μ, σ*) = (2πσ2)-1/2 exp (-(x-p?/(2σ2)) exp (-(x-μ)2 a) How many parameters does a mixture of m Gaussians have? b) Let xi, , Vn be n observations drawn from a mixture of m Gaussians. Write down the log-likelihood function. Hint: it should involve two summations c) Let 1 k < m. Show that the maximum likelihood estimator for μk is given by rn where d) Let 1 < k < m. Show that the maximurn likelihood estimator for σん is given by rn where ?ki is as defined above

please work out parts b,c,d with clear steps thanks

Answer 1