MA 590  Homework 9 

MA 590 
Homework 9 
Problems
1 (20 points) This problem is a prelude to statistical estimators, which we will discuss from
the Bayesian perspective as we move forward in class. Consider the Rayleigh distribution
X ∼ Rayleigh(σ
2
)
with pdf given by
π(x) = x
σ
2
exp 
−
x
2
2σ
2

, x ≥ 0
as derived in Homework 8.
(a) Unlike the normal distribution, the maximizer of π(x) for the Rayleigh distribution does
not coincide with its expected value. Verify this by computing the maximizer of π(x) and
the expected value of the Rayleigh distribution, and generate a plot to show your results
graphically. In computing the maximizer, it may help to note that
arg max
x
π(x) = arg min
x
V (x), V (x) = − log(π(x)).
(b) Assume we have N independent and identically distributed (i.i.d.) random variables X1, . . . , XN ,
such that
Xj ∼ π(x | θ), j = 1, . . . , N,
where the pdf of Xj depends on the value of a parameter θ.
In frequentist statistics, the Maximum Likelihood Estimator (MLE) of the parameter θ is
the value of θ that maximizes the probability of the outcomes xj
, j = 1, . . . , N; i.e.,
θML = arg max
θ
π(x1, . . . , xN | θ) = arg max
θ
Y
N
j=1
π(xj
| θ)
provided such a maximizer exists.
If X1, . . . , XN are i.i.d. with
Xj ∼ Rayleigh(σ
2
), j = 1, . . . , N,
compute the MLE of θ = σ
2
.
2 (10 points) Provide a brief update on your final project, describing the progress you’ve
made over the past few weeks. What have you done thus far? Have you thought about
some of the considerations mentioned in the comments on your Final Project Proposals in
Canvas? (If you haven’t already replied, please outline your replies to my comments here!)
Please keep in mind that we will have synchronous final project presentations via Zoom
during our class time 3:00-4:50 PM on Monday, May 11, and Wednesday, May 13 – more
details to follow!
Note: For any of the above problems for which you use MATLAB to help you solve, you must
submit your code/.m-files as part of your work. Your code must run in order to receive full credit.
If you include any plots, make sure that each has a title, axis labels, and readable font size, and
include the final version of your plots as well as the code used to generate them.
2