Solution to Exercise C7
Let X Geometric(p). Recall that
EX =
1
2p
and EX 2 =
.
p
p2
EX 3 =
=
=
=
=
E (X 3 | X = 1)P (X = 1) + E (X 3 | X > 1)P (X > 1)
1 p + E (1 + (X 1)3 | X > 1 (1 p)
1 p + E (1 + X
Joint density of Order Statistics
Suppose X1 , X2 , . . . , Xn are iid with pdf f (x).
Let (U1 , U2 , . . . , Un ) = (X(1) , X(2) , . . . , X(n) ).
Then
fU1 ,U2 ,.,Un (u1 , u2 , . . . , un )
= n! f (u
For a discrete random variable X and event A, dene
E (X | A) =
x P (X = x | A)
xX
E (g (X ) | A) =
g (x)P (X = x | A)
xX
whenever the sums converge absolutely.
Analog of Law of Total Probability for e
Suppose X P , .
Basus Lemma: If T (X ) is complete and sucient (for ),
and S (X ) is ancillary, then S (X ) and T (X ) are independent for
all .
In other words, a complete sucient statistic is indepen
(Probabilistic) Experiment: (, B, P )
is the sample space set of all possible outcomes.
(Often denoted S.)
denotes a particular outcome.
= cfw_ all possible
B is the class of events for which prob
http:/www.stat.fsu.edu/~huffer/mordor/5327/test3_material/ch8.homewor.
Test #3 will be on Wednesday, April 22.
Exercise: Find the Fisher information matrix for a k-parameter
exponential family with th
http:/www.stat.fsu.edu/~huffer/mordor/5327/test2_material/ch7_hw_pro.
Reading:
All of Ch. 7 except 7.2.4 (EM algorithm)
Background reading: Section 5.3, pages 218-220
and Section 5.5, pages 232-236
an
Assignment #1
Note: Homework exercises are not handed in. However, homework is strongly stressed
on the exams. Most of the exam problems will be similar to homework exercises.
Reading: All of Chapter
Miscellaneous Solutions and Comments on the Chapter 7 Exercises
7.4: For X1 , . . . , Xn iid N (, 1), the likelihood function may be found by setting = and 2 = 1 in the expression for the joint densit
Further Details on Exercise 7.9
= X(n) so that nxn1 n n+1 n n n x dx = n = dx = n n 0 n+1 n+1 0 n1 n n+2 n n n+1 2 nx 2 2 = x E x fX(n) (x) dx = x dx = n = 2 dx = n n 0 n+2 n+2 0 2 2 n n n Var() = .