MAST20004-14-Assign4.pdf

MAST20004 Probability — 2014 Assignment 4

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, 30 May. There Th ere are 6 quest questio ions, ns, of wh whic ich h 2 rando randoml mly y chos chosen en quest questio ions ns will will be mark marked ed (chosen (chosen after assignment submission). 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  Let  X  have have a normal distribution distribution with E [X ] = 0 and Var Var (X ) =  σ 2 for a σ a  σ >  0. (a) Find the moment moment generating generating function function of  Y  =  X 2 , making sure to determine the domain where it is defined. (b) If  X 1 , . . . , Xn   are independent and have the same distribution as X , find the moment generating function of Z  of  Z  =  X 12 + + X n2 .

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(c) Hence show show Z   Z   has a gamma gamma distrib distributio ution n and ident identify ify its paramete parameters. rs. [This distribution is called a “chi-squared” distribution and is important in statistics.] statistics.] 2. At a certain coffee cart, there are on average 400 paying customers per day with a standard standard deviation deviation of 10. The averag averagee a custome customerr spends spends is $5 with a standard deviation of $2. (a) If we assume the amount amount each customer spends is independent independent of the number of customers in a day, find the average revenue per day for the cart. (b) If in addition to the assumption in part (a), we also assume the amounts that the customers spend are all independent of each other, then what is the standard deviation of the revenue per day for the coffee cart? 3. Let X   X   have an exponential distribution with rate 1 and the distribution of  Y  X  =  x  be Poisson with parameter x.

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(a) What is the expectation of Y  of  Y ?? (b) What is the variance of Y  of  Y ?? 1

(c) Find the probability generating function of  Y   and use it to identify the distribution of Y   by name. 4. Let X   be exponential with rate one. (a) Show that E [X a−1 ] = Γ(a) for any a > 0. (b) Using a change of variable and then relating to the normal distribution density, show that Γ(1/2) = π. (c) By using a three term Taylor series approximation for ψ(x) = x−1/2 , find an approximation for π by evaluating E [ψ(X )]. How good is the approximation? Think about how you could make the approximation better. (d) Use a two term Taylor’s series approximation to approximate Var (ψ(X )). How good is this approximation?

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5. Let U   be uniform on the interval (0, 1) independent of  V   which has density f V  (v) = 2(1 v) for 0 < v