Please answers fast and correct i will definetly upvote.

This question uses a probability distribution called the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter if

P(X|)= ^Xe/X! .

The Poisson distribution is a useful discrete distribution that can be used to model the number of occurrences of something per unit of time. For example, if a bank teller sits at a counter, the number of customers arriving in each interval, say 30 minutes, is in the Poisson distribution.

Here, we will estimate the parameter from n observations {X1,...,Xi,...Xn} (e.g., the number of customers for the ith teller in 30 minutes) which we assume are drawn i.i.d from the Poisson distribution.

1. (3 points) Compute the log-likelihood for observations {X1, . . . , Xn}.

2. (4 points) Compute the MLE for .

3. (8 points) Now lets be Bayesian and put a prior distribution over the parameter . Your extensive experience in statistics tells you that the good prior distribution for is a Gamma distribution.

p(|,) = ()1e, > 0.

It is well known that the mode of in the Gamma distribution with and is ( 1)/ for > 1. Recall that the mode in the statistics represents the value that appears most often (i.e., the maxima of the probability mass function). Then, compute the MAP for . (Hint: PXi+1e(n+) can be represented by a Gamma distribution.)