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Bayesian learning and Monte Carlo
Simulations
Homework 1
April 2020
A researcher collects data about electrical engineering students and he is
interested by estimating the proportion, p, of the number of students that study
less than 5 hours per day. Our experimental sample of size 1000 gives us 648
students that study less than 5 hours.
1 Exercise
a. What is the probability distribution of the data? Compute the likelihood
and plot it. Note: take everything in percentage. Add a line of the sample
proportion to the plot. (Hint: use x = seq(1, 100, 1), size = 100 as
parameters in the cumulative distribution).
b. Which continuous probability distribution should be used to describe the
prior of this proportion? Specify the function of R and the support.
2 Exercise
Another researcher claimed that only 40% of students study less than 5 hours,
and this with a variance of 0.2. We want to take this information as a prior for
our study. How can we do that? Represent this graphically.
3 Exercise
Find the posterior distribution of p. Then, plot it together with the prior on
the same graph. What do you notice?
4 Exercise
What are the 95% credible region using HPD and using quantiles? Plot them
together with the posterior on the same graph. (Hint: try a sequence from 3 to
6 by 0.1 for h).
1
5 Exercise
What would happen if we observe a sample of size 10 instead of 1000? And we
observe 6 statisticians that study less than 5 hours.
2