Homework 8
Due Thursday Nov 29 by 3:00
1. Let X1 , . . . , Xn Bernoulli(p). Let p =
1
sample and let p = n n=1 Xi .
i
n
i=1
1
n
Xi . Let X1 , . . . , Xn denote a bootstrap
(a) What is the distribution of np , conditional on X1 , . . . , Xn ?
(b) Find an e
LIST OF THEOREMS
Theorem 3:
Let G be a group.
(i) If x a = x b or a x = b x, then a = b.
(ii) The identity element e is unique.
(iii) For all x G, the inverse element x1 is unique.
1
(iv) For all x G we have (x1 ) = x.
(v) For all a, b G we have (a b)1 =
GROUPS: SUPPLEMENTARY TOPICS
Theorem (see Proposition 2.59, page 172):
If G is a nite abelian group, then G has a subgroup of order d for every divisor d of |G|.
Theorem (Cayley) (see Theorem 2.66, page 178):
Every group G is isomorphic to a subgroup of t
GROUPS
Denition 1:
An operation on a set G is a function : G G G.
Denition 2:
A group is a set G which is equipped with an operation and a special element e G, called
the identity, such that
(i) the associative law holds: for every x, y, z G,
x (y z ) = (
GROUPS
Denition 1:
An operation on a set G is a function : G G G.
Denition 2:
A group is a set G which is equipped with an operation and a special element e G, called
the identity, such that
(i) the associative law holds: for every x, y, z G,
x (y z ) = (
GROUPS
Denition:
An operation on a set G is a function : G G G.
Denition:
A group is a set G which is equipped with an operation and a special element e G, called
the identity, such that
(i) the associative law holds: for every x, y, z G,
x (y z ) = (x y
Homework 3
36-705
Due: Thursday Sept 27 by 3:00
1. Let C = A
B . Show that
sn (C ) sn (A) + sn (B ).
2. Let C = cfw_A B ; A A, B B. Show that
sn (C ) sn (A)sn (B ).
3. Show that sn+m (A) sn (A)sm (A).
4. Let
A=
A = cfw_a [b, c] cfw_d : a b c d .
Find VC d
Homework 6
36-705
Due: Thursday November 1 by 3:00
1. 10.31 (a,b,c,e)
2. Show that, when H0 is true, then the p-value has a Uniform (0,1) distribution.
3. Let X1 , . . . , Xn N (, 2 ) where both and 2 are unknown.
(a) Find the Wald test for
H0 : = 0
versu
Permutations
Denition:
A permutation of a set X is a rearrangement of its elements.
Example:
1. Let X = cfw_1, 2. Then there are 2 permutations:
12, 21.
2. Let X = cfw_1, 2, 3. Then there are 6 permutations:
123, 132, 213, 231, 312, 321.
3. Let X = cfw_1,
LAGRANGES THEOREM
Denition:
An operation on a set G is a function : G G G.
Denition:
A group is a set G which is equipped with an operation and a special element e G, called
the identity, such that
(i) the associative law holds: for every x, y, z G we hav
ISOMORPHISMS OF GROUPS
Denition:
If (G, ) and (H, ) are groups, then a function f : G H is a homomorphism if
f (x y ) = f (x) f (y )
for all x, y G.
Example:
Let (G, ) be an arbitrary group and H = cfw_e, then the function f : G H such that
f (x) = e for
HOMOMORPHISMS AND ISOMORPHISMS
Denition:
If (G, ) and (H, ) are groups, then a function f : G H is a homomorphism if
f (x y ) = f (x) f (y )
for all x, y G.
Example:
Let (G, ) be an arbitrary group and H = cfw_e, then the function f : G H such that
f (x)
Groups
Denition:
An operation on a set G is a function : G G G.
Denition:
A group is a set G which is equipped with an operation and a special element e G, called
the identity, such that
(i) the associative law holds: for every x, y, z G,
x (y z ) = (x y
Homework 8
Due Thursday Nov 15 by 3:00
1. Let X1 , . . . , Xn P where P has a density p on [0, 1]. Assume that the density p has
a bounded, continuous derivative p . Given an integer m, let h = 1/m and dene m
bins:
B1 = [0, h], B2 = (h, 2h], . . . .
Let n
COMMUTATIVE RINGS
Denition:
A commutative ring R is a set with two operations, addition and multiplication, such that:
(i) R is an abelian group under addition;
(ii) ab = ba for all a, b R (commutative law);
(iii) a(bc) = (ab)c for any a, b, c R (associat
Homework 2
36-705
Due: Thursday Sept 20 by 3:00
1. Let X1 , . . . , Xn be independent random variables. We do not assume that they are
identically distributed. Let i = E(Xi ). Assume that ai Xi bi for i = 1, . . . , n.
Prove an exponential bound for
P
1
n