MATH5905 Term One 2025 Assignment One Statistical Inference
University of New South Wales
School of Mathematics and Statistics
MATH5905 Statistical Inference
Term One 2025
Assignment One
Given: Friday 28 February 2025 Due date: Sunday 16 March 2025
Instructions: This assignment is to be completed collaboratively by a group of at most
3 students. Every effort should be made to join or initiate a group. (Only in a case that you
were unable to join a group you can present it as an individual assignment.) The same mark
will be awarded to each student within the group, unless I have good reasons to believe that a
group member did not contribute appropriately. This assignment must be submitted no later
than 11:59 pm on Sunday, 16 March 2025. The first page of the submitted PDF should be
this page. Only one of the group members should submit the PDF file on Moodle, with the
names, student numbers and signatures of the other students in the group clearly
indicated on this cover page. By signing this page you declare that:
I/We declare that this assessment item is my/our own work, except where acknowledged, and
has not been submitted for academic credit elsewhere. I/We acknowledge that the assessor of
this item may, for the purpose of assessing this item reproduce this assessment item and provide
a copy to another member of the University; and/or communicate a copy of this assessment
item to a plagiarism checking service (which may then retain a copy of the assessment item on
its database for the purpose of future plagiarism checking). I/We certify that I/We have read
and understood the University Rules in respect of Student Academic Misconduct.
Name Student No. Signature Date
1
MATH5905 Term One 2025 Assignment One Statistical Inference
Problem One
a) Suppose that the X and Y are two components of a continuous random vector with a density
fX,Y (x, y) = 12xy
3, 0 < x < y, 0 < y < c
(and zero else). Here c is unknown.
i) Find c.
ii) Find the marginal density fX(x) and FX(x).
iii) Find the marginal density fY (y) and FY (y).
iv) Find the conditional density fY |X(y|x).
v) Find the conditional expected value a(x) = E(Y |X = x).
Make sure that you show your working and do not forget to always specify the support of
the respective distribution.
b) In the zoom meeting problem from the lecture, show that the probability that if there
are 40 participants in the meeting then the chance that two or more share the same birthday,
is very close to 90 percent.
Problem Two
A certain river floods every year. Suppose that the low-water mark is set at 1 and the high-
water mark X has a distribution function
FX(x) = P (X ≤ x) = 1− 1
x3
, 1 ≤ x <∞
1. Verify that FX(x) is a cumulative distribution function
2. Find the density fX(x) (specify it on the whole real axis)
3. If the (same) low-water mark is reset at 0 and we use a unit of measurement that is 110
of that used previously, express the random variable Z for the new measurement as a
function of X. Find the cumulative distribution function and the density of Z.
Problem Three
a) A machine learning model is trained to classify emails as spam or not spam based on certain
features. The probability that an email is spam is 0.3. The probability that the model predicts
spam given that the email is actually spam, is 0.9. The probability that the model predicts
spam given that the email is not spam, is 0.15. If a randomly received email is classified as
spam by the model, what is the probability that the email is actually spam?
b) In a Bayesian estimation problem, we sample n i.i.d. observations X = (X1, X2, . . . , Xn)
from a population with conditional distribution of each single observation being the geometric
distribution
fX1|Θ(x|θ) = θx(1− θ), x = 0, 1, 2, . . . ; 0 < θ < 1.
The parameter θ is considered as random in the interval Θ = (0, 1) and is interpreted as a
probability of success in a success-failure experiment.
i) Interpret in words the conditional distribution of the random variable X1 given Θ = θ.
2
MATH5905 Term One 2025 Assignment One Statistical Inference
ii) If the prior on Θ is given by τ(θ) = 30θ4(1 − θ), 0 < θ < 1, show that the posterior
distribution h(θ|X = (x1, x2, . . . , xn)) is also in the Beta family. Hence determine the Bayes
estimator of θ with respect to quadratic loss.
Hint: For α > 0 and β > 0 the beta function B(α, β) =
∫ 1
0 x
α−1(1 − x)β−1dx satisfies
B(α, β) = Γ(α)Γ(β)Γ(α+β) where Γ(α) =
∫∞
0 exp(−x)xα−1dx. A Beta (α, β) distributed random vari-
able X has a density f(x) = 1B(α,β)x
α−1(1− x)β−1, 0 < x < 1, with E(X) = α/(α+ β).
iii) Seven observations form this distribution were obtained: 2, 3, 5, 3, 5, 4, 2. Using zero-one
loss, what is your decision when testing H0 : θ ≤ 0.80 against H1 : θ > 0.80. (You may use the
integrate function in R or any favourite programming package to answer the question.)
Problem Four
A manager of a large fund has to make a decision about investing or not investing in certain
company stock based on its potential long-term profitability. He uses two independent advi-
sory teams with teams of experts. Each team should provide him with an opinion about the
profitability. The random outcome X represents the number of teams recommending investing
in the stock to their belief (based on their belief in its profitability).
If the investment is not made and the stock is not profitable, or when the investment is
made and the stock turns out profitable, nothing is lost. In the manager’s own judgement, if
the stock turns out to be not profitable and decision is made to invest in it, the loss is equal
to four times the cost of not investing when the stock turns out profitable.
The two independent expert teams have a history of forecasting the profitability as follows. If
a stock is profitable, each team will independently forecast profitability with probability 5/6
(and no profitability with 1/6). On the other hand, if the stock is not profitable, then each
team predicts profitability with probability 1/2. The fund manager will listen to both teams
and then make his decisions based on the random outcome X.
a) There are two possible actions in the action space A = {a0, a1} where action a0 is to
invest and action a1 is not to invest. There are two states of nature Θ = {θ0, θ1} where
θ0 = 0 represents “profitable stock” and θ1 = 1 represents “stock not profitable”. Define
the appropriate loss function L(θ, a) for this problem.
b) Compute the probability mass function (pmf) for X under both states of nature.
c) The complete list of all the non-randomized decisions rules D based on x is given by:
d1 d2 d3 d4 d5 d6 d7 d8
x = 0 a0 a1 a0 a1 a0 a1 a0 a1
x = 1 a0 a0 a1 a1 a0 a0 a1 a1
x = 2 a0 a0 a0 a0 a1 a1 a1 a1
For the set of non-randomized decision rules D compute the corresponding risk points.
d) Find the minimax rule(s) among the non-randomized rules in D.
e) Sketch the risk set of all randomized rules D generated by the set of rules in D. You
might want to use R (or your favorite programming language) to make the sketch precise.
3
MATH5905 Term One 2025 Assignment One Statistical Inference
f) Suppose there are two decisions rules d and d′. The decision d strictly dominates d′ if
R(θ, d) ≤ R(θ, d′) for all values of θ and R(θ, d) < (θ, d′) for at least one value θ. Hence,
given a choice between d and d′ we would always prefer to use d. Any decision rule that is
strictly dominated by another decisions rule is said to be inadmissible. Correspondingly, if
a decision rule d is not strictly dominated by any other decision rule then it is admissible.
Indicate on the risk plot the set of randomized decisions rules that correspond to the
fund manager’s admissible decision rules.
g) Find the risk point of the minimax rule in the set of randomized decision rules D and
determine its minimax risk. Compare the two minimax risks of the minimax decision
rule in D and in D. Comment.
h) Define the minimax rule in the set D in terms of rules in D.
i) For which prior on {θ1, θ2} is the minimax rule in the set D also a Bayes rule?
j) Prior to listening to the two teams, the fund manager believes that the stock will be
profitable with probability 1/2. Find the Bayes rule and the Bayes risk with respect to
his prior.
k) For a small positive ϵ = 0.1, illustrate on the risk set the risk points of all rules which
are ϵ-minimax.
Problem Five
The length of life T of a computer chip is a continuous non-negative random variable T with
a finite expected value E(T ). The survival function is defined as S(t) = P (T > t).
a) Prove that for the expected value it holds: E(T ) =
∫∞
0 S(t)dt.
b) The hazard function hT (t) associated with T is hT (t) = limη→0
P (t≤Tη .
(In other words, hT (t) describes the rate of change of the probability that the chip survives
a little past time t given that it survives to time t.)
i) Denoting by FT (t) and fT (t) the cdf and the density of T respectively, show that
hT (t) =
fT (t)
1− FT (t) = −
d
dt
log(1− FT (t)) = − d
dt
log(S(t)).
ii) Prove that S(t) = e−
∫ t
0 hT (x)dx.
iii) Verify that the hazard function is a constant when T is exponentially distributed,
i.e., fT (t) = βe
−tβ, t > 0 where β > 0 implies that hT (t) = β.
iv) Let now T be Pareto distributed with parameter θ > 0, i.e., with a density
fT (t|θ) = θ
tθ+1
when t ≥ 1
(and zero else). Find its hazard function.
4

学霸联盟