STAT2010: Fundamentals of Statistics Assignment 3 2013 - SOLUTIONS
STAT2010: Fundamentals of Statistics
Assignment 3 2013 - SOLUTIONS
This assignment is to be submitted at the beginning of the 3pm to 5pm lecture on Thursday October 31, 2013 (Week 12).
Assignments may be hand written but must be neat, legible and appropriately formatted. Assignments, including a standard cover page, are to be on A4 paper secured in the top left hand corner by (preferably) a staple.
Solutions should be written in clear English with all appropriate working and/or supporting computer work shown.
This is an individual assignment, NOT a group assignment. If you work collectively with other students on this assignment you may be penalised for plagiarism. See the course outline for more information.
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Question 1
a) Consider a continuous random variable X with a domain of -∞ < x < ∞. Show
that for a constant c, if Y = X + c, then Var(Y) = Var(X). (11 marks)
b) Suppose that a series of independent and identically distributed (iid) random
variables Xi has an exponential distribution with parameter
l
. Show that S =
X1 + X2 + … + Xk has an Erlang distribution with parameters k andl .
(9 marks)
Question 2
Let X and Y have the joint pdf
fXY(x, y)= x2+y
A sketch of the pdf is given below.
(45 marks)
0£ x£1, 0£ y£1
Sketch of the pdf | f | XY(x, y)=x | 2 | + | y,0£x£1, 0£y£1 | |
a)Fine the marginal pdfs, f | X | (x) and fY(y) |
b) Comment on whether X and Y are independent?
c)Show that | E(Xr Ys)= | 1 | + | 2 | ||
(r+ 3)(s+1) | (r+1)(2s+3) | |||||
d) From your result in c), findmX andmY
e) Determine sX andsY
(10 marks)
(3 marks)
(7 marks)
(10 marks)
(8 marks)
f) Find the correlation between X and Y. How does this result compare with your
finding in part b)?
Question 3
The zero truncated Poisson distribution has probability function given by
x | ||||
P(X= x)= | l | |||
(e | l | -1)x! | ||
for x = 1, 2, …, in whichl > 0.
Given a random sample X1, ..., Xn, show that the method of moments estimator and the maximum likelihood estimator coincide and both satisfy
ˆ | ||||||
X= | l | ˆ . | ||||
1- e | -l | |||||
Note. You will need to show that E[X] = | l | -l . You may assume, just on this | ||||
1 | - e | |||||
occasion, that the stationary point of the logarithm of the | likelihood | gives the | |
universal maximum. | |||
Question 4 | (16 marks) | ||
a)Show that if the random variable X has | moment | generating | function |
MX(t) = E[etX] then Y = aX has moment generating function MX(at). (3 marks)
b) The random variable X has the gamma (a,l) distribution if, for – l < t <l, it has moment generating function
M | -a | ||
X | (t)={1-(t /l)} | ||
.
Show that if X1, …, Xn is a random sample from the gamma (a,l) distribution
then
X
has the gamma (na, nl) distribution.
(7 marks)
c) It follows in part from the result in b) that since a gamma (a,l) distribution has
meana/l and variancea/l2,X approximately has the N(a/l,a/(nl2)) distribution. Use this information to construct a 95% two-sided confidence interval forl when X1, …, Xn is a random sample from the gamma (a,l) distribution witha known. (6 marks)
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Solution: STAT2010: Fundamentals of Statistics Assignment 3 2013 - SOLUTIONS