STAT 2011 Probability and Estimation Theory – Semester 1, 2023
Computer Practical Sheet Week 8
Computer Problems for Week 8
For this week’s lab report: Please submit your code, output, and any comment
required, for Q3 ONLY : the pdf or html file must include code, output and any
comment, otherwise you will be penalised. An assignment item has been set up on
Canvas; file upload format is limited to pdf and html.
1. Reconsider the Week 8 tutorial problem 9: Let X and Y have the joint pdf fX,Y (x, y) =
2e?(x+y), 0 < x < y, 0 < y. Confirm P (Y < 3X) = 1/2 using Monte Carlo Integration
with the number of uniformly distributed points J ranging over 100, 1000 and 10000
and restricting Y to be less than 10. The main challenges will be in finding out how to
generate uniform (pseudo)random numbers over the (restricted) region Ry<10 = {(x, y) :
0 < y < 3x < 10}, how to calculate V (R) and ensuring that fX,Y (x, y) = 0 whenever
0 < x < y and 0 < y does not hold.
(Hint: Generate uniform (pseudo) random numbers (x, y) from [0, c1] × [0, c2] and then
only use those that satisfy y < 3x and x < y. Check with plot(x,y) if the sampled
points look uniform over the desired region.)
2. We will visualise the joint cdf and pdf for two continuous random variables X and Y .
The joint cdf is defined as FX,Y (x, y) =
1
3x
2(2y + y2) for 0 ≤ x, y ≤ 1.
(a) Generate a grid of x, y values on the interval [0, 1]:
x = seq(0, 1, length= 30)
y = x
(b) Define a function f equal to the joint cdf, and plot the surface.
f = function(x, y) { z = x^2*(2*y+y^2)/3 }
z = outer(x, y, f)
persp(x,y,z,theta=45,col = "lightblue")
(c) Repeat the above for the joint pdf, fX,Y (x, y) =
4
3x(1 + y), 0 ≤ x, y ≤ 1.
1
3. Consider two RVs, X ～ Bin(m, p), Y ～ Bin(n, p), with m = 10, n = 20, p = 1/3. We
investigate the distribution of the sum W = X + Y , in particular how close the empirical
probability distribution (obtained via simulation) comes to the theoretical probability
distribution Bin(n+m, p).
(a) Set B=100. Generate a random sample of size B from Bin(m, p) and store the result
in X. Similarly, generate a random sample of size B from Bin(n, p) and store the
result in Y. Compute W and tabulate the values using table(W).
(b) Convert the column names of the table to numeric, store in W.obs, and compute the
empirical frequencies of different values of W , store in P.obs:
Tab=table(W)
W.obs = as.numeric(names(Tab))
P.obs = as.numeric(Tab/B)
(c) Plot the empirical frequencies and the theoretical probabilities from Bin(n +m, p)
(obtained using dbinom(0:(m+n),m+n,p=1/3)), and comment on how close they are.
(d) Repeat for B = 10, 000. Comment.