Suppose we observe random variable $Y_i$inR associated with an unobserved label $C_i$inM,$W,$

Assuming both classes share a known variance ${}^2,$ but different unknown means ${}_M,{}_N.$ The

distribution for each class is

$P(c_i)=q^{1(C_i)}=M(1-q{)}^{1(c_i)}=W$

The $q$ is know, and the conditional distributions with each class follow a Gaussian:

$p(y_i|c_i)=pro{d}_{c}N(y_i$;${}_i,{}^2{)}^{(c_i)}=c$

Compute the posterior probability distribution for $C_i$ given the observations

Compute the expected joint probability for $Y$ and $C,$ and then show differentiate w$.$r$.$t$.{}_M$

and ${}_W.$