Solution to Exercise C7
Let X Geometric(p). Recall that
and EX 2 =
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 )3 (1 p)
1 p + E 1 + 3X + 3X 2 + X 3 (1 p)
p + 1 + 3EX
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) ).
fU1 ,U2 ,.,Un (u1 , u2 , . . . , un )
= n! f (u1 )f (u2 ) f (un )I (u1 < u2 < < un ) .
The order stati
For a discrete random variable X and event A, dene
E (X | A) =
x P (X = x | A)
E (g (X ) | A) =
g (x)P (X = x | A)
whenever the sums converge absolutely.
Analog of Law of Total Probability for expectations:
If the events B1 , B2 , . . . , Bk form a
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
In other words, a complete sucient statistic is independent
of any ancillary statistic.
Preliminary remarks (p
(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 probabilities are dened.
(We mainly ignore B in this class.
Test #3 will be on Wednesday, April 22.
Exercise: Find the Fisher information matrix for a k-parameter
exponential family with the natural parameter w(theta)=theta.
Read Sections 8.1,
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 1 and Section 2.1.
A1, A2, A3 (do these rst
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 density given in Example 6.2.7 on page 277. This leads to L(|
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() = . 2 = n+2 n+1 (n + 2) (n + 1)2 E = xfX(n) (x) dx = x MSE