Sampler method question

Let (X1, Z1), ..., (Xn, Zn) be i.i.d. pairs, where each Z; is a Bernoulli random variable with P(Z; = 1) = a and P(Z; = 2) = 1 - a for a E [0, 1]. Moreover, let the conditional distribution of X; given Z; = z be Exponential with rate 1 > 0 when z = 1 and rate 12 > 0 when z = 2. Note that the marginal distribution of X; is the mixture distribution a . Exponential()1) + (1 -a) . Exponential(12). 1. Write the log-likelihood function based on the complete data (X1, Z1), ..., (X,, Z,) in terms of the statistics n n n i=1 i=1 2. Find the maximum likelihood estimator (MLE) of (a, )1, 12) when observing both X1, ..., X, and Z1, ..., Zn. 3. Derive an EM algorithm for computing the MLE of (a, )1, 12) when observing only X1, ..., Xn. 4. Derive a gradient descent algorithm for computing the MLE of (a, )1, 12) when observing only X1 , . .., Xn