School of Mathematics and Statistics
Te Kura Mātai Tatauranga
DATA304/COMP312/DATA474
Assignment 1
Due 17 March, 18:00
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Problem #1 (3 marks)
Consider n independent identically distributed random variables X1 , X2 , . . . , Xn which obey the Bernoulli
distribution with probability of success Pr(Xi = 1) = p, with 0 ≤ p ≤ 1. What is the distribution of
U = min{X1 , X2 , . . . , Xn }? What is the distribution of V = max{X1 , X2 , . . . , Xn }?
Problem #2 (2 marks)
How would you describe, using the Kendall-Lee notation, a system in which packages arrive at regular
times, e.g., one every five seconds, they are checked by four customer service representatives (CSR) that
take a random time to verify their contents (this time may be assumed exponentially distributed with
mean η > 0, and the CSR do not communicate). Every now and then, packages arrive with a red tag;
red-tagged packages are immediately opened by any of the four CSR. The maximum line capacity is of
one hundred packages, and they arrive from the whole world.
Problem #3 (2 marks)
Still referring to the previous system, assume all CSR are busy. What is the expected time for the first
red-tagged package in the line to be opened? What is the expected time for the first red-tagged package
in the line to be checked?
Problem #4 (3 marks)
Consider the Erlang distribution characterized by the following probability density function:
fZ (z) =
θk
z k−1 e−θz 1R+ (z),
(k − 1)!
where θ, z > 0 and k ∈ N (the natural numbers excluding zero). Find a parametrization for this density
that uses the mean µ = E(Z). Show the code that implements this probability density function. Compute
the variance using this parametrization. Show plots of this reparametrized density with µ = 3 and varying
values of the variance.
DATA304/COMP312/DATA474, 2023
1
Assignment 1