This assignment will strengthen your understanding of the fundamental concepts in applied
probability and statistics. The questions in this assignment will build on lecture material as
well as the assigned readings from (Wasserman, L. 2004. \All of Statistics”).

Deliverables Submit your responses as a PDF along with any code as separate files. Show
partial credit if the work demonstrates understanding.

Problem 1: Maximum Likelihood Estimation (2 points)

I would like to build a simple model to predict how many students are likely to come to
my office hours this semester. Because this is an arrival process, I will model the number
of arrivals during office hours as Poisson distributed. Recall that the Poisson is a discrete
distribution over the number of arrivals (or events) in a fixed time-frame. The Poisson
distribution has a probability mass function (PMF) of the form,

a) During my last three office hours I received X1 = 10;X2 = 11;X3 = 8 students. Write
the logarithm of the joint probability distribution log p(X1;X2;X3; ).

b) Compute the maximum likelihood estimate (MLE) of the rate parameter MLE which max-
imizes the joint probability in part (a). The model is concave and so the MLE can be com-
puted by nding the zero-derivative solution. Make sure to show all of your calculations.
How many arrivals should I expect at my next office hours under this model?

c) I have assigned a particularly challenging homework which has led to a lot of students
X4 = 25 arriving at my office hours. Compute the MLE again, but include this new
training point. How has the model changed with this new data?

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