MAST20004 Probability — 2014
Assignment 3

If you didn’t already hand in a completed and signed Plagiarism Declaration Form (available from the LMS or the department’s webpage), please do so and attach it to the front of this assignment.
Assignment boxes are located on the ground floor in the Richard Berry Building (north entrance). Your solutions to the assignment should be left in the MAST20004 assignment box set up for your tutorial group. Don’t forget to staple your solutions and to print your name, student ID, the subject name and code, and your tutor’s name on the first page (not doing so will forfeit marks). The submission deadline is 5pm on Friday, 9 May.
There are 6 questions, of which 2 randomly chosen questions will be marked (chosen after assignment submission). Note you are expected to submit answers to all questions, otherwise a mark penalty will apply. Give clear and concise explanations. Clarity, neatness and style count.

1. Let X be uniform on the set {1/2,1/3,1/4}, and let the distribution of Y |X = x be geometric with parameter x; that is, P(Y = k|X = x) = (1 - x)kx, for k = 0,1,2,....
(a) What is the probability mass function of Y ?
(b) What is the mean of Y ?
(c) Find the probability mass function of X|Y = k for k = 0,1,2,...
2. A company that manufactures cogs sells them in cartons of 100. It is historicallyknown that about 1% of the cogs manufactured by the company are defective.
(a) Write an exact expression for the probability that a carton has more than2 defective cogs in it.
(b) Approximate this same probability using the normal distribution.
(c) Approximate this same probability using the Poisson distribution.
(d) Which of the approximations of parts (b) and (c) is better? Why is thisthe case?
3. Assume that Q is a random variable with density proportional to q for 0 q 1. Given Q = q, N has a binomial distribution with parameters n and q. The following identity for a,b -1 is useful for this problem:
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(a) What is the probability mass function of N?
(b) What is the mean of N and the expected value of 1/(N + 1)?
(c) Find the density of Q|N = k for k = 0,1,2,...,n
4. The lifetime of a certain type of electrical component is well modeled by an exponential distribution with mean 2 hours. Assume that a system uses one component of this type and when there is a failure, the component is immediately replaced by another of the same type with exponential lifetime independent of all previous components.
(a) Given the first component has not failed after two hours, what is the chanceit will not fail within the following two hours?
(b) What is the chance that at least two components fail in the first hour?
5. Assume that X and Y are independent exponential random variables with rates 1 and 2, respectively.
(a) Find the distribution function of the maximum of X and Y .
(b) Find the density of the minimum of X and Y . Can you recognise the distribution of the minimum by name?
(c) What is the probability that X Y ?
6. Let (X,Y ) be a point uniformly chosen on the unit disc {(x,y) : x2 + y2 1}.
(a) Write down the density of (X,Y ).
(b) What is the density of the angle 0 T 2p made between the positive x-axis and the ray connecting the origin to the point (X,Y )?
(c) What is the density of X?
(d) What is the density of Y |X = x for -1 x 1?
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