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
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 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 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 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
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 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
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
The University of Sydney
STAT2911 Probability and Statistical Models
Semester 1
Computer Class Week 12
2013
Use RStudio to get your lab report. Your report is due by Friday 4pm Week 12.
We will use MC simulations and parametric bootstrap to investigate th
STAT2911
Probability and Statistical Models
Tutorial 11
1. Show that the joint density of (U, V ) = (X(k1) , X(k) ) is given by
fU V (u, v) =
n!
f (u)f (v) [F (u)]k2 [1 F (v)]nk
(k 2)!(n k)!
u v.
2. Show that for a uniform (0, 1) distribution and Rk = X(k
STAT2911 Lab - Week 11
May 19, 2013
1. The more obvious way to transform the U (0, 1) sample into a N (0, 1) sample is to apply the inverse of
the N (0, 1) CDF, 1 : qnorm(U)
2. > box_muller = function(U) cfw_
+
R = sqrt(-2*log(1-U[1])
+
W = 2*pi*U[2]
+
re
STAT2911
Probability and Statistical Models
Tutorial 11 Solutions
1. We will give rst the intuitive argument. The innitesimal probability that U = Xk1
(u, u + du) and V = Xk (v, v + dv) and Xi Xk1 for i = 1, . . . , k 2, Xi Xk for
i = k + 1, . . . , n is
STAT2911
Probability and Statistical Models
Tutorial 12
1. Find the method of moments estimators of the parameters and for a shifted exponential sample x1 , , xn , from a distribution with density
f (x) = e(x) ,
x .
2. (Rice Chapter 4 Q13) If X is a non-n
STAT2911
Probability and Statistical Models
Tutorial 12 Solutions
1. Let Y be an exponential(1) RV, i.e., fY (y) = ey 1(0,) (y). The density of X = Y / +
is given by
fX (x) = fY (x ) = e(x) 1(0,) (x ) = e(x) 1(,) (x).
Therefore,
E(Y )
1
+ =+
V (Y )
1
V (
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
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 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
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 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 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
The University of Sydney
STAT2911 Probability and Statistical Models
Semester 1
Computer Class Week 11
2013
Use RStudio to get your lab report. Your report is due by Friday 4pm Week 11.
The Box-Muller method allows us to generate a sample of size 2 from t
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 9 Solutions
1. Let U be the point where the line is cut, then U U (0, 1). The longer piece is more
than twice the length of the shorter piece i U < 1/3 or U > 2/3 so the probability that
will happen is
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 3 Solutions
1.
(i) Let X, Y and Z = n X Y be the numbers of white, black and red drawn. Then
for 0 k, l, k + l n
P (X = k, Y = l, Z = n k l) = P (X = k, Y = l) =
w
k
b
r
l nkl
w+b+r
n
.
(ii)
P (X + Y =