ST2133 ZA
Summer 2021 Online Assessment Instructions
ST2133 Advanced statistics: distribution theory
Friday, 28 May 2021: 15:00 - 19:00 (BST)
The assessment will be an open-book take-home online assessment within a 4-hour window. The requirements for this assessment remain the same as the closed- book exam, with an expected time/effort of 2 hours.
Candidates should answer all FOUR questions: QUESTION 1 of Section A (40 marks) and all THREE questions from Section B (60 marks in total). Candidates are strongly advised to divide their time accordingly.
You should complete this paper using pen and paper. Please use BLACK INK only.
Handwritten work then needs to be scanned, converted to PDF and then uploaded to the VLE as ONE individual file including the coversheet. Each scanned sheet should have your candidate number written clearly in the header. Please do not write your name anywhere on your submission.
You have until 19:00 (BST) on Friday, 28 May 2021 to upload your file into the VLE submission portal. However, you are advised not to leave your submission to the last minute.
Workings should be submitted for all questions requiring calculations. Any necessary assumptions introduced in answering a question are to be stated.
You may use any calculator for any appropriate calculations, but you may not use any computer software to obtain solutions. Credit will only be given if all workings are shown.
If you think there is any information missing or any error in any question, then you should indicate this but proceed to answer the question stating any assumptions you have made.
The assessment has been designed with a duration of 4 hours to provide a more flexible window in which to complete the assessment and to appropriately test the course learning outcomes. As an open-book exam, the expected amount of effort required to complete all questions and upload your answers during this window is no more than 2 hours. Organise your time well.
You are assured that there will be no benefit in you going beyond the expected 2 hours of effort. Your assessment has been carefully designed to help you show what you have learned in the hours allocated.
Section A
Answer all three parts of question 1 (40 marks in total)
1. (a) The probability mass function of a random variable X is given by
pX (x) = cqx-1, x = 2, 3, · · · .
where 0 < q < 1 and c is a constant.
i. Find the value of c. [4 marks]
ii. Find P (X > 2.5) and P (X is even). Do not leave your answer in terms of c [5 marks]
iii. Find the moment generating function of X . Do not leave your answer in terms of c. [5 marks]
(b) Let X follows the standard normal distribution.
i. Show that the moment generating function of X is given by
MX (t) = et2/2, t ∈ R. [5 marks]
ii. State the Markov inequality for a non-negative random variable Y. [1 mark]
iii. Show that, for t > 0,
[6 marks]
(c) In a game, a fair die is thrown independently n times. Let X be the total number of throws showing 3 or higher. If X ≤ 1, you lose the game.
i. Show that
[4 marks]
ii. Suppose m independent games are played. Write down the probability mass function of Y , where Y denotes the number of games lost. [2 marks]
iii. Now suppose that the number of games played, M, follows a Poisson distri- bution with mean μ (so the answer to part ii. is the conditional mass function of Y given M = m). Show that
[8 marks]
Section B
Answer all three questions in this section (60 marks in total)
2. The joint probability density for the random variables X and Y is given by
(a) Show that c = 1/(log 3 — log 2). [4 marks]
(b) Define the transformation
i. Show that the joint density fU,V (u, v) of U and V is given by
You should show clearly how you arrive at the region on the (u, v) plane where the density is defined. [8 marks]
ii. Work out the marginal density of V. [2 marks]
iii. Find
(Hint: You can find first) [6 marks]
3. Let X1 , X2 , . . ., be a sequence of independent and identically distributed random variables. Let N be Poisson distributed with mean μ and is independent of the Xi’s. Define
We define W = 0 if N = 0.
(a) Suppose each Xi has density
fX (x) = λe-λx, x > 0.
Work out the moment generating function for W given N. [4 marks]
(b) Show that the moment generating function of W is given by
[5 marks]
(c) Calculate the mean and variance of W. [5 marks]
(d) Now consider Z = NX1 . Find the mean and variance of Z. [6 marks]
4. A county is made up of three (mutually exclusive) communities A, B and C, with proportions of them given by the following table:
|
Community
|
A
|
B
|
C
|
|
Proportion
|
0.2
|
0.5
|
0.3
|
Given a person belonging to a certain community, the chance of that person being vaccinated is given by the following table:
|
Community given
|
A
|
B
|
C
|
|
Chance of being vaccinated
|
0.8
|
0.7
|
0.6
|
(a) We choose a person from the county at random. What is the probability that the person is not vaccinated? [5 marks]
(b) We choose a person from the county at random. Find the probability that the person is in community A given the person is vaccinated. [4 marks]
(c) If a person is vaccinated, the probability that they eventually show symptoms is 0.1, while the probability is 0.6 for a non-vaccinated person. For a person who eventually shows symptoms, the waiting time, T , until the symptoms appear is exponentially distributed, with rate 1/4 if they are vaccinated (i.e. T ~ Exp(1/4)), and rate 1/2 if they are not.
i. Given that a person eventually shows symptoms, but these symptoms have not yet appeared at time T = 2, find the probability that this person is not vaccinated. [6 marks]
ii. Given symptoms are shown eventually, find the mean of T. [5 marks]