February 27, 2013
STAT 244 Solution # 7
STAT 244: Homework Solution # 7
Xiang Zhu
Total points: 130
Problem 8.10.7
(b) [10 pts] The log likelihood is
n
p(1 p)xi 1 = n log p + n(n 1) log(1 p)
x
l(p) = log
i=1
where xn =
1
n
n
i=1 xi
is the sample mean. To
February 12, 2013
STAT 244 Solution # 5
STAT 244: Homework Solution # 5
Xiang Zhu
Total points: 125 + 20
Problem 3.8.42
(a) [10 pts] The cdf of X is
FX (x) = P (X x|W = 1)P (W = 1) + P (X x|W = 1)P (W = 1)
1
1
= P (T x) + P (T x)
2
2
11
= + (P (T x) P (T
HW4 solutions
Stat 244 Fall 2009
Takintayo Akinbiyi
2.5.67
a) To get the density f from the cumulutive distribution function F we dierentiate F .
fX (x) =
1
x
d
F (x) =
dx
x
exp
, x>0
b) For x > 0
P (X x)
=P
W
x
1
= P W x
1
= F (x )
=
1 exp
x
1
1 ex
d
January 18, 2013
STAT 244 Solution # 1
STAT 244: Solution # 1
Xiang Zhu
Total points: 105
Problem 1.8.6
(a) [5 pts] Observe that P (A) + P (B ) + P (C ) counts each of P (A B C ), P (B C A), P (C A B )
twice, and counts P (A B C ) three times. The negativ
February 5, 2013
STAT 244 Solution # 3
STAT 244: Homework Solution # 3
Xiang Zhu
Total points: 130
Problem 2.5.2
The sample space
= cfw_HHHH, HHHT, HHT H, HHT T, HT HH, HT HT, HT T H, HT T T,
T HHH, T HHT, T HT H, T HT T, T T HH, T T HT, T T T H, T T T T
Stat244 FAll2009 HW8
Takintayo Akinbiyi
9.11.2
a) Simple hypothesis
b) Simple hypothesis
c) Composite hypothesis
d) Composite hypothesis
8 marks, 2 each
9.11.3
a)
=
P (|X 50| > 10; p = .5)
=
P
X 100p
100p(1 p)
>
100p(1 p)
; p = .5
P (|Z | > 2) Z N (0, 1)
February 12, 2013
STAT 244 Problem Session # 4
STAT 244: Problem Session # 4
Xiang Zhu
1
Discussion Problems
1.1
Rice 3. 8. 61, 62
If X and Y are independent standard normal random variables. Find:
an expression for the joint density of U = a + bX and V
February 12, 2013
STAT 244 Problem Session # 5
STAT 244: Problem Session # 5
Xiang Zhu
1
Review
1.1
Important Concepts
expected values/mean of discrete/continuous random variables; variance; standard deviation; mean squared
error; covariance; correlation;
Stat244 HW2 Solutions
Oct 14 2009
Takintayo Akinbiyi
Throughout note AB = A B
1.8.54
a)
=
c
c
P (Ri+1 |Ri )P (Ri ) + P (Ri+1 |Ri )P (Ri )
=
p + (1 )(1 p)
=
P (Ri+1 )
p( + 1) + (1 )
b)
=
c
c
P (Ri+2 |Ri+1 )P (Ri+1 ) + P (Ri+2 |Ri+1 )P (Ri+1 )
=
P (Ri+1 )(
January 21, 2013
STAT 244 Problem Session # 2
STAT 244: Problem Session # 2
Xiang Zhu
1
Review
1.1
Important Concepts
discrete random variable; probability mass function/frequency function; cumulative distribution function
(cdf) and its properties; indepe
January 28, 2013
STAT 244 Problem Session # 3
STAT 244: Problem Session # 3
Xiang Zhu
1
Review
1.1
Important Concepts
density function; uniform distribution/density; quantile; exponential distribution/density(memoryless);
gamma distibution/density(shape p
March 18, 2013
STAT 244 Midterm
Statistics 24400 Midterm Solution
Xiang Zhu
1. Solution:
Notice that transform y = g (u) = log (1/u) is continuous and monotone on interval (0, 1). The
inverse of g is g 1 (y ) = ey for y (0, ). Hence, the pdf of Y is
1
g
STAT 244 HW6 FALL 2009
Takintayo Akinbiyi
5.4.2
P (|X n | > )
=
P (|(X n n ) + (n )| > )
P (|X n n | + |n | > )
P (|X n n | > /2 |n | > /2)
P (|X n n | > /2) + P (|n | > /2)
=
P (|X n n | > /2) + 1(|n | > /2)
Use Markov Inequality
V (X n )
+ 1(|n | > /2)
Statistics 244 : Homework 5
Solution
Winter 2013
Page 1
Problem 4.7.36(Partial)
Let Ai (i = 1, 2) be the event in which i th half tests positive at the rst stage. Similarly, let Bj
(j = 1, 2, 3, 4) be the event in which j th quarter tests positive at the