Construct an em algorithm for model

Question

Solve the following problem:

Consider the sample x = (0.12, 0.17, 0.32, 0.56, 0.98, 1.03, 1.10, 1.18, 1.23, 1.67, 1.68, 2.33), generated from an exponential mixture

p Exp(λ) + (1 - p) Exp(μ).

All parameters p, μ, λ are unknown.

a. Show that the likelihood h(p, λ, μ) can be expressed as E[H(x, Z)], where z = (z₁,...,z₁₂) corresponds to the vector of allocations of the observations xi to the first and second components of the mixture; that is, for i = 1,..., 12,

 P(z_i = 1) = 1 - P(z_i = 2 = pλexp(-λx_i)/ pλexp(-λx_i) + (1 - p)μ exp(-μ x_i)

b. Construct an EM algorithm for this model, and derive the maximum likelihood estimators of the parameters for the sample provided above.