MATH50013 Probability and Statistics for JMC, Autumn 2022
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MATH50013 Probability and Statistics for JMC, Autumn 2022
Coursework
Due on Friday, December 2nd at 13:00 GMT
Instructions:
● This coursework will be collected via Turnitin at Blackboard > Course Content > Assessments > Coursework 1 Drop Box Autumn 22.
● Your Coursework Submission should be entitled CID_Coursework1. Nowhere on the submission should you use your name.
● You may hand-write solutions if your handwriting is good. A typed report is better.
● To receive marks you must show your work and/or explain your reasoning. Show how you worked out the solution rather than just give the final expression.
● You cannot discuss the problems with your classmates or anyone else.
● If you have questions of clarification do not hesitate to ask me, but please ask privately (either on the discussion forum or by email) rather than as a public post. If needed, I will post clarifications to Blackboard and send an announcement.
Other information:
● This coursework is worth 10% of the total module mark.
● All marks are provisional and subject to moderation by the exam board.
1. Let X be a continuous random variable with support X and PDF of the form
f (x) = h(x) exp{t(x)θ 一 A(θ)}, x e x , (1)
for some functions h(x), t(x), and A(θ), where θ > 0 is a parameter.
(a) Assuming that t(x) is bijective, show that the PDF of t(X) can be written in the
form (1), for some other functions 
, 
, and 
. [5 points]
(b) Show that E(exp{ut(X)}) = exp{A(θ + u) 一 A(θ)} for all u > 一θ .
(c) Let Y be a continuous random variable with support (0, o) and PDF [5 points]
f (y) = ′ 
exp ,一 θ(y2y(一) 1)2 } , y > 0,
where θ > 0. Show that the PDF of Y can be written in the form (1), indicating the functions h(y), t(y), and A(θ). [5 points]
(d) Find the mean of exp{一(Y + Y一1 )/2}. [5 points]
2. Let (X, Y) be a random vector such that X ~ Exponential(λ) and the conditional dis- tribution of Y given X is Y I X = x ~ Gamma(α, x). In other words, its conditional
density is given by
fY |X(y I x) = 
yα 一1 exp{一xy}, y > 0.
(a) Find the marginal PDF of Y . [20 points]
(b) Show that X I Y = y follows a Gamma distribution and find its parameters. [20 points]
3. Let X1 , X2 , . . . , Xn be independent random variables, where Xi has CDF Fi (x) and PDF fi (x) for x e R.
(a) Derive from first principles the CDFs and PDFs of the smallest and largest values in the samples, M = min{Xi : i = 1, . . . , n} and Z = max{Xi : i = 1, . . . , n}, expressed as functions of the CDFs and PDFs of the Xi ’s. [10 points]
(b) Assume that Xi ~ Exponential(λi ). What distribution does M follow? Explain this result intuitively using the interpretation of the exponential distribution as the time between events in a Poisson process. [10 points]
(c) In the same setting as (b) with λi = λ for all i = 1, . . . , n, show that P (Z < z) → 0 as n → o for all z > 0, and find a sequence z1 , z2 , . . . such that P (Z < zn ) → exp(一x) as n → o, for some fixed x > 0. [5 points]
(d) Suppose that a computer program needs to perform n different tasks, and that the tasks are sent to a cluster to be run in parallel. Assume that the time to completion of all the tasks follows an Exponential(λ) distribution, and that they run independently. The main program finishes when all the tasks finish. Find the average time we need to wait for the main program to finish. [15 points]
2023-01-08