STAT2911
Probability and Statistical Models
Tutorial 6 Solutions
1.
(i) Recall that E(X) = m/2 therefore m = 2E(X) and m = 2.
x
(ii) The likelihood is
L(m) =
(
(m + 1)
0
n
0 x 1 , , xn m
.
otherwise
This is maximized for m = maxcfw_x1 , , xn .
(iii) E(M )
STAT2911
Probability and Statistical Models
Tutorial 6
1. Let X be a discrete uniform RV on 0, 1, , m; that is
P (X = k) = 1/(m + 1)
k = 0, 1, . . . , m.
(i) Find the moment estimate of m based on a sample x1 , , xn .
(ii) Find the maximum likelihood esti
The University of Sydney
STAT2911 Probability and Statistical Models
Semester 1
Computer Class Week 6
2013
Use RStudio to get your lab report. Your report is due by Friday 4pm Week 6.
We will examine the relative accuracy of estimation by the method of mo
STAT2911
Probability and Statistical Models
Tutorial 2 Solutions
1.
(i) We must choose a value for 3 and 2 other values then for the rst value we choose
3 suits and for the others a suit each out of all possible deals of 5 cards:
13
12
4
4
4
52
/
2
3
1
1
STAT2911
Probability and Statistical Models
Tutorial 10
1. Let X and Y be independent exponential (1) rv. Find the joint distribution of Z = X +Y
and W = X/Y . Hence show that Z and W are independent and give the marginal
densities.
2. Let X and Y have jo
STAT2911
Probability and Statistical Models
Tutorial 4 Solutions
1. Note that for t > 0,
X x tX tx etX etx .
Therefore
P (X x) = P etX etx .
Let Y = etX then Y 0 so we can use Markovs inequality: for y > 0,
P (Y y)
E(Y )
.
y
Choose y = etx to conclude th
STAT2911
Probability and Statistical Models
Tutorial 9
1. (Rice 2.5.37) A line segment of length 1 is cut once at random. What is the probability
that the longer piece is more than twice the length of the shorter piece?
2. (Rice 2.5.59) If U is uniform on
STAT2911
Probability and Statistical Models
Tutorial 7
1. Let X be a Poisson RV and nd E(1/(X + 1). Use this to show that in general
E(X/Y ) = E(X)/E(Y ).
2.
(i) Simplify E(X|X) = . . . .
(ii) Assume X and Y are independent and E(X) exists. Find E(X|Y ).
STAT2911
Probability and Statistical Models
Tutorial 5 Solutions
1. Recall that for a multinomial random vector N = (N1 , . . . , Nr ) with parameters (m ; p1 , . . . , pr ),
Ni Binom(m, pi ). Therefore E(Ni ) = mpi , and COV (Ni , Ni ) = V (Ni ) = mpi (1
The University of Sydney
STAT2911 Probability and Statistical Models
Semester 1
Computer Class Week 4
2013
Use RStudio to generate your lab report. Your report is due by Friday 4pm Week 4.
This week we simulate the sampling of balls from an urn with and w
STAT2911
Probability and Statistical Models
Tutorial 8
1. (Rice 4.7.74) The number of ospring of an organism is a discrete RV with mean and
variance 2 . Each of its ospring reproduces in the same manner (and independently of
all other osprings). Find the
STAT2911
Probability and Statistical Models
Tutorial 8 Solutions
1. Let N be the number of osprings in the second generation and let Xi be the number
of osprings the ith organism in the second generation has. Then the number of osprings in the third gener
STAT2032/6046 - Solutions to Additional Questions Lecture Week 4
1. A company will receive a perpetuity of $50,000 per annum. The first
payment under this agreement is due at t = 10. What is the present value
of this agreement at t = 0, assuming an annual
STAT2032/6046 - Solutions to Additional Questions Lecture Week 6
1. Sam borrows $500,000 at an annual effective rate of 5% pa interest for
25 years. Monthly repayments of $X are made at the end of every month.
After 5 years, the bank increases the interes
STAT2032/6046 - Solutions to Additional Questions Lecture Week 5
1. For a price of $100, you receive monthly interest payments of $1 in
arrears for 5 years and a lump sum payment of $X at the end of 5 years.
Assuming an effective annual rate of interest o
STAT2911
Probability and Statistical Models
Tutorial 10 Solutions
1.
fX,Y (x, y) = e(x+y) , x > 0, y > 0.
Let Z = X + Y , W = X/Y . Put z = g1 (x, y) = x + y, w = g2 (x, y) = x/y. Then the
inverse transformation is x = h1 (z, w) = zw/(1 + w), y = h2 (z, w
STAT2032/6046 - Solutions to Additional Questions Lecture Week 2
1. What deposit made today will provide for a payment of $1,000 in 1 year and
$2,000 in 3 years, if the effective compound rate of interest is 7.5% pa?
The deposit made is the present value
STAT2911 Lab Week 2 Solutions
Longen Lan 440351423
March 11, 2015
1
Binomial and Poisson Distributions
This week, the computer lab tutorial explores the circumstances and how a Binomial distribution can be
approximated as a Poisson Distributions by answer
STAT2911 Lab Week 1
Longen Lan 440351423
March 3, 2016
1
Old Faithful Geyser In Yellowstone National Park
The following report concerns the data involving the duration of eruption and the waiting time between eruptions
at the famous Old Faithful Geyser in
STAT2911 Lab Week 5 Solutions
Longen Lan 440351423
April 1, 2015
1
Estimation by Method of Moments
Today we explore why we estimate parameters by the lowest possible moment when we estimate using the
method of moments.
2
Solutions
1. We first set a sample
STAT2911 Lab Week 3 Solutions
Longen Lan 440351423
March 18, 2015
1
Expectation and Variance
Today we explore expectation and variance of random variables, as covered in the lectures.
Solutions
1. Define a sequence vector to plot the Binom(10, 0.3) barplo
STAT2911 Lab Week 4 Solutions
Longen Lan 440351423
March 25, 2015
1
Drawing balls form an urn
Today we explore sampling balls from an urn with and without replacement with the motivation
being to estimate the proportion of red balls in the urn.
2
Solution
STAT2911 Lab Week 6 Solutions
Longen Lan 440351423
April 15, 2015
1
Esimation: MOM v MLE
Today we explore the relative accuracy of estimation by the method of moments (MOM) versus the maximum
r k
likelihood estimate (MLE). The example we will use is a ne
The University of Sydney
The School of Mathematics and Statistics
Assignment 2 Solutions
Longen Lan 440351423
STAT2911: Probability and Statistical Models (Advanced)
Lecturer: Uri Keich
Semester 1, 2015
Any result stated in class or given as a tutorial pr
STAT2911 Lab - Week 3
March 15, 2013
1)
>
>
>
>
n = 10
p = 0.3
ks = c(0:n)
bin_pmf = dbinom(ks, n, p)
The graph will be plotted below with all the other graphs.
2)
> E = sum(ks * bin_pmf)
> V = sum( (ks - E)^2 * bin_pmf )
> E
[1] 3
> V
[1] 2.1
3)
>
>
>