Stat 426 Winter 2015
Homework 2 Solutions
Takeaways from this homework.
Q1a: How do we find of mode for a discrete distribution?
Q2a: Not only the range of values of the random variable, but also the range of possible
values of its parameters affects prob
Homework 4
Review lectures 4 & 5, Sec 2.3, 3.7, 4.5
Review question 67, Chapter 2; question 69, Chapter 3; 79, 81, 82, 83, 84 Chap 4
1. Let f (x) = x1, x1, > 0, and0otherwise
.
a. Find F (x)
x
x
F (x) = y1dy = y|1 = 1 x
1
b. How can we generate random va
Stat 426 Exam 1 Solution F15
Problem 1 (Show all reasoning)
15 points
Consider a simplified model for weather forecasting. Let = cfw_ and let =
( ). Further, suppose that the probability that the weather (either rain or no rain) tomorrow
will be the same
'32: MH- Hmr) 43,: -=é.,.;.fl
3. gafffgg img (.4
{1L +§Menl~ if
, (ward cam! . fl/gag
" dikb-A x 4-f H
w; WM M433 MVWW we m am, A _
. a -; a . _ 7: . r a; gum,
" ~ f \gw- Maw 3, I _ l
g I (: f, (@hbaubwLComf-qlrlj acids}
' = San-Pk 5.1%.; 3 Wm M 4 )
m
Home 6 I
{law/{2 2 aLom;
'* EKFeoMh
_._ . ' " no a." 26:; 3"
lh IMF/4 M LW'Vivg My {Mfg ) Efgjgarrf-n
~._. Familial: r viva/10W; Wrp'a I'FS Why.
_._ Max/m
#004,41an WM: [EU/(y) a s a {35145.1 912 x,
am HM pwjm; 5(acrzx)J:X,-~
9% I
E W) W?-
f+ (WW1 t
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Review problems
1. Let (X1 , Y1 ), . . . , (Xn , Yn ) be i.i.d. with density
f (x, y ) =
2/2 ,
0
x > 0, y > 0, x + y <
otherwise.
Find a 1-dimensional sucie
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 6
Issued: Thursday, October 11, 2012
Due: In class Tuesday, October 23, 2012
Reading: Rice Chapter 6, Chapter 8: Sections 14
Do the following pro
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 8
Issued: Thursday, November 1, 2012
Due: In class Thursday, November 8, 2012
Reading: Chapter 8.5.1, and 8.5.2
Note: As we will see in the next
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 7
Issued: Thursday, October 18, 2012
Due: In class Thursday, October 25, 2012
Reading: Chapter 8: Sections 15.1
Do the following problems from Ch
Solutions for HW12
9.2 Hypotheses (a) and (b) are simple; (c) and (d) are not. The reason (d) is not is that
X N (0, 2 ) where 2 can be anything 0.
9.4 Ordering xi according to their likelihood ratio (x), we get
x
x 4 x2 x1 x3
f0 ( x )
3
(x) := f1 (x) 1 4
Solutions for HW11
Important note: Unless otherwise stated, x = (x1 , . . . , xn ) and X = (X1 , . . . , Xn ).
8.4 (e) Let x01 = i 1cfw_xi cfw_0, 1, that is the number of data points that are either 0 or 1.
Similarly, x23 = i 1cfw_xi cfw_2, 3. Verify that
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 9
Issued: Thursday, November 8, 2012
Due: In class Thursday, November 15, 2012
Reading: Sections 8.5 amd 8.7
Do the following problems from Chapt
Stat 426 : Homework 4.
Moulinath Banerjee
November 8, 2015
Announcement: The homework carries a total of 60 points (10 6).
(1) A level 1 C.I. for the normal mean , when is assumed known, is given by:
X n q(1 2 , N (0, 1) , X n q(1 , N (0, 1) ,
n
n
where
Homework 6 Stat 426 Fall 2015 Solutions
Problem 1 (Question 3, Chapter 6). Let be the average of a sample of 16 independent normal
random variables with mean 0 and variance 1. Determine c such that P(| < c) = .5. Use: qnorm(p)
to compute quantiles. H
Homework 5 Stat 426 Solutions
Review Lecture 6. Practice problems: Q 1, 7, 8, 9, 13, 15 Chap 5
1. Let " , $ , be a sequence of independent random variables with ( ) = (
"
"
and ( ) = ($ . Show that if as , 4 ( and 6 4 ($ 0,
(5
(5
"
4
Stat 426 W2015
Homework 1 Solution
Problem 1
A lie detector test has a 90% detection rate if a person is lying, and a 5% detection rate if a
person is not lying. Suppose that 1% of all individuals in a cert
V0,? OF 0.5 MIDFAQ normal/31'; 137/ MlE:
Pzall M m+n4+on jam) #7 {mm
1F +u zzMalfme is mm, M MLE n w saws" or
I} ,HL (Xi) :0
Ag r,v.,.L.'./e «F \_':'_\/\J
hawk- can 4% Tm)
Aka le-I' %(1):Zo(7). Than, wL [Am/z
- _| h .
xynm Tigm»
Li'! 6 W2 know NF"
7%) M
We would. VL [)/h" [Mum/W41 5"]? fowl
{)4 ) X2} "7qu
[Tel
27 94 WW ZI'MNOIU
we cm Jake as our IDS! {WM m 57u4&/{ wb/ 10/940:-
Exawflt. L. 44% Famous ZWPIL 1 on. cam aka ynoM +51 Vat/[ma 01/ +141
V1019, MS an man IDAfM-LLM 61 .1
X; r 44, 05" , i, 1- Mom
T
Camuwlri MH' SWIM Mm PM! Swfa vafima.
PM WM 96447 - Awnmncrympn)
W We 5W7 MA 5 = i. gxl.
W- (pm ML)
>7 N (X.§)(1'Yr"/wa m WWW.
Cm. )7 MA 52 are nlLTalm.
pmp. 114.2, a Mme gm 3 i: a #Auan M (x,_)?,.,)<n_32) MIN-CL ,1 (17,4 04);
nltM/ng. (h_l)$z~x1[.
J" "
STAT 426
Lecture 34
Fall 2012
Arash A. Amini
September 13, 2012
1 / 35
Announcements
My oce hours:
Tue 4 5p in 470 West Hall,
Wed 12 1p in 438 West Hall
Yingchuans oce hours:
Wed 2:30 3:30p in 274 West Hall
Fri 9:30 10:30a in 274 West Hall
Final exam: Wed
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 12
Issued: Thursday, December 6, 2012
Not to be turned in
Reading: Sections 9.1, 9.2 and 9.4 in 3rd edition
Do the following problems from Chapte
U of Michigan
Department of Statistics
STAT 426: Introduction to theoretical statistics
Fall 2012
Problem Set 11
Issued: Thursday, November 29, 2012
Due: In class Friday, December 6, 2012
Reading: Sections 8.6 in 3rd edition
Do the following problems from
STAT 426
Lecture 20
Fall 2012
Arash A. Amini
November 20, 2012
1 / 37
Outline
Suciency
What does it really mean?
Sucient statistic is not unique
Minimal sucient
Order statistics are sucient for iid data
Exponential families
More examples
Some properties
M