Instructions: Questions labelled with ‘(R)’ require use of R. Please provide appropriate
R commands and their output, along with sufficient explanation and interpretation of the
output to demonstrate your understanding. Such R output should be presented in
an integrated form together with your explanations. All other questions should be
completed without reference to any R commands or output, except for looking up quantiles
of distributions where necessary. Make sure you give enough explanation so your tutor can
follow your reasoning if you happen to make a mistake. Please also try to be as succinct as
possible. Each assignment will include marks for good presentation.
1. Let X1; : : : ; Xn be a random sample from the Log-normal distribution, LN(µ; σ), whose
Hint: you may use the fact that if Z ∼ N(0; 1), then X = exp(µ+σZ) ∼ LN(µ; σ).
Also, the moment generating function (MGF) of a normal random variable may
be useful here.
(b) Show that the method of moments estimator (MME) of µ and σ are
(c) Show that the maximum likelihood estimator (MLE) of µ and σ are
2. (R) The daily new COVID-19 cases were surging in Victoria since last Christmas. Let
X be a random variable representing the number of new COVID-19 cases reported in
Victoria. The following are 16 observations of X (from December 25, 2021 to January
2108, 1608, 1999, 2738, 3767, 5137, 5919, 7442,
7172, 8577, 14020, 17636, 21997, 21728, 51356, 44155
(a) Give basic summary statistics for these data and produce a box plot. Briefly
comment on center, spread and shape of the distribution.
(b) Assuming a Log-normal distribution (LN(µ; σ) as in Question 1), compute maxi
mum likelihood estimates for the parameters.
(c) Draw a density histogram and superimpose a pdf for a Log-normal distribution
using the estimated parameters.
(d) Draw a QQ plot to compare the data against the fitted Log-normal distribution.
Include a reference line. Comment on the fit of the model to the data. Hint: Quan
tile for the Log-normal distribution may be computed using the qlnorm function
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