STAT 2006 Assignment 2
Due Time and Date: 3 p.m., 21st Feb, 2014
1. Let X1 , X2 , . . . , Xn be random samples from bin(1, p) where 0 < p < 1.
(a) Show that X =
n
i=1
Xi
is an unbiased estimator of p.
n
p(1 p)
(b) Show that Var X =
.
n
X(1 X)
p(1 p)
(c) S
THE CHINESE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS
STAT 2006 Basic Concepts in Statistics and Probability II
MID-SEMESTER TEST 2013 - 2014
24th Feburary, 2014
Time allowed: One and a half hour.
Name:
Student No.:
Curriculum:
Candidates may use a
STAT 2006 Tutorial 10
8th/10th/11th April 2013
Non-Parametric Method: Chi-Square Goodness-of-Fit Tests
1. Statistical inference on the distribution modeling concerning whether our proposed distribution
for an experiment is correct or not
2. In general, le
STAT2006 Tutorial 11
15th/17th/18th Apr 2013
1. Cramr-Rao Lower Bound
Let X = (X1 , X2 , ., Xn ) be a random sample from a distribution of the continuous type with p.d.f.
f (x; ), where the support of X does not depend upon . Let l(x; ) be the log-likelih
STAT 2006 Tutorial 12 Solution
22th/24th/25th April 2013
1. X = 50; Y = 110; = 2; = 110 2(50) = 10; 2 = 6. Therefore, the tted linear regression is
iid
Yi = 10 + 2xi + i where i N (0, 6).
95% two-sided C.I. for : [1.891694315,2.108305685]
95% two-sided
STAT 2006 Tutorial 12
22th/24th/25th April 2013
1. Linear Regression
Standard Model: Y = + X + where Y, X, are called dependent variable, independent variable
and random error term, respectively. is the intercept and is the slope. In short, assume Y and
STAT 2006 Tutorial 1
21st/23rd/24th Jan 2013
Sample space and probability;
Random variables, discrete random variables and their distributions.
1. Sample space, outcome and event
A sample space, usually denoted by S, is just a set.
An outcome, usually
STAT2006 Tutorial 2
28th/30th/31st Jan 2013
Continuous random variables and their distributions.
Covariance and correlation
1. Continuous random variables, c.d.f. and p.d.f.
Continuous random variable - random variables with uncountable (must have inni
STAT2006 Tutorial 6
4th/6th/7th, Mar 2013
1.
(a)
p=
388
y
=
= 0.38
n
1022
(b)
z0.05 = 1.65, n = 1022,
The approximate 90% condence interval for p is
y
z/2
n
(y/n)(1 y/n) y
, + z/2
n
n
= [0.38 1.65
(y/n)(1 y/n)
n
0.38(1 0.38)
, 0.38 + 1.65
1022
0.38(1 0.3
STAT 2006 Tutorial 10
8th/10th/11th April 2013
Non-Parametric Method: Chi-Square Goodness-of-Fit Tests
1. Statistical inference on the distribution modeling concerning whether our proposed distribution
for an experiment is correct or not
2. In general, le
STAT2006 Tutorial 6
4th/6th/7th, Mar 2013
I Condence intervals for proportions
(1) condence interval for p
Y B(n, p), let Y /n be an estimator of p.
Y np
np(1 p)
=
(Y /n) p
p(1 p)/n
has an approximate normal distribution N (0, 1). Then we obtain
p z/2
(Y
STAT 2006 Tutorial 7
11th/13th/14th Mar 2013
1. Linear Regression (Cancelled)
Standard Model: Y = + X + where Y, X, are called dependent variable, independent variable
and random error term, respectively. is the intercept and is the slope. In short, assu
STAT2006 Tutorial 7 Solution
11th/13th/14th Mar 2013
1. Cancelled X = 50; Y = 110; = 2; = 110 2(50) = 10; 2 = 6. Therefore, the tted linear
iid
regression is Yi = 10 + 2xi + i where i N (0, 6).
95% two-sided C.I. for : [1.891694315,2.108305685]
95% two-
STAT2006 Tutorial 8 Solution
18th/20th/21st Mar 2013
1. (a) Test statistic is T =
X7.5
,
S/ 10
and the critical region is C = cfw_X : |t| t0.025 (9).
(b) x = 7.55, s = 0.103, t0.025 (9) = 2.262
7.557.5
For t = 0.103/10 = 1.532 < 2.262, we do not reject H
STAT2006 Tutorial 9
25th/27th/28th, Mar 2013
1. In each of the following situations, calculate the p-value of the observed
data.
(a) For testing H0 : 1/2 versus H1 : > 1/2, 8 successes are out of 10
Bernoulli trials.
(b) For testing H0 : 1 versus H1 : > 1
STAT2006 Tutorial 8
18th/20th/21st Mar 2013
1. Tests about One Mean
Null hypothesis H0 : = 0 .
Three possibilities for alternative hypothesis:
(i) > 0 ;
(ii) < 0 ;
(iii) = 0 .
When the variance is known, consider the test statistic,
Z=
X 0
,
/ n
which i
(a)
STATZDOG Tutorial 11 Solution
Note that
a g Var[tY + Z] = Var[Y]t2 -:- 200t[y, Z]t + W712] V75 6 it
Note that the RHS is a quadratic expression in t (WLOG we assumed Var[Y] > 0). A quadratic
expression is nonnegative only if its discriminant is non-po
STAT 2006 Tutorial 1
21st/23rd/24th Jan 2013
Sample space and probability;
Random variables, discrete random variables and their distributions.
1. Sample space, outcome and event
A sample space, usually denoted by S, is just a set.
An outcome, usually
STAT2006 Tutorial 3
4th/6th/7th, Feb 2013
1. X =
U +V
2
,Y =
U V
2
J=
, u, v R.
X
U
Y
V
X
V
Y
V
=
1
2
1
2
Thus, fU,V (u, v) = fX,Y ( u+v , uv )|J| =
2
2
1 1 (u2 +v 2 )
e 4
, u, v
4
1
2
1
2
1
2
=
exp 1
2
1
2
u+v 2
2
+
uv 2
2
1
2
=
R. For U, V are bivariat
STAT2006 Tutorial 3
4th/6th/7th, Feb 2013
1. Variable transformation
Suppose random variables X1 , X2 , ., Xn have joint pdf fX1 ,X2 ,.,Xn and
we have variable transformations Yi = ui (X1 , X2 , ., Xn ), i = 1, 2, ., n. We
want to nd the joint distributio
STAT 2006 Assignment 1 Suggested Solution
1. (a)
0
(1 FX (x)dx
= [x(1
FX (x)]
0
FX (x)dx
0
0
xd(1 FX (x)
[xFX (x)]0
xfX (x)dx =
0
(b)
0
xfX (x)dx +
=
xdFX (x)
+
0
xfX (x)dx = E[X].
The Jacobian approach: Note Z = X + Y, W = X X = W, Y = Z W .
Hence th
STAT 2006 Assignment 2 Suggested Solution
i.i.d.
1. Let X1 , X2 , . . . , Xn Bin(1, p) where 0 < p < 1.
n
i=1
(a) E X = E
Xi
=
n
n
i=1
(b) Var X = Var
Xi
n
n
i=1
E [Xi ]
np
=
= p. Therefore, X is an unbiased estimator of p.
n
n
=
n
i=1
n2
Var (
Xi )
n
i=1
STAT2006 Assignment 3
Due Date: 15:00, 24th March, 2015
1. (a) Let Y be an exponential random variable with mean and X 1 + 2 Y, 2 > 0. Find the pdf
of X and remember to state the support of X. X is said to follow a shifted exponential
distribution with lo
STAT 2006 Assignment 4
Due Date: 16th April, 2015
1. Let Y be Bi(100,p). To test H0 : p = 0.08 against H1 : P < 0.08, we reject H0 if and only if Y<=6.
(a) Determine the signicance level of the test
(b) Calculate the power function if in fact p=0.04.
2. T
1
Week 8
MASTER BUDGET AND
RESPONSIBILITY
ACCOUNTING
106/04/15
Definition of Budget
2
A budget is
The quantitative expression of a proposed
plan of action by management for a
specified period, and
An aid to coordinating what needs to be
done to implemen
STAT 2006 Tutorial 8
28th/29th/31th Mar 2017
1. Condence interval for the dierence of two means
2
Let X1 , X2 , ., Xn be a random sample from N (X , X
) and Y1 , Y2 , ., Ym be another random
2
sample from N (Y , Y ), and the two samples are independent.
STAT 2006 Assignment 1
Due Time and Date: 5 p.m., 8th Feb, 2017
1. Assume Town A has three parties, Party B, Party C and Party D, with n1 , n2 and n3 citizens in
them, respectively. Assume, for people in B, they will always support the motion by the
gover