STAT 517
Practice FinalFall 2012
Name:
Please return this page with your solution after exam.
1. Let Y1 < Y2 denote the order statistics of a random sample of size 2 from N (0, 2 ).
(a). Show that EY1 = / .
(b). Find the covariance of Y1 and Y2 .
2. Let t
Notation and Terminology used on Exam MLC
Version: March 1, 2012
In actuarial practice there is notation and terminology that varies by country, by application, and
by source. The purpose of this study note is to present notation and terminology that will
S UFFICIENCY
A Sufcient Statistics for a Parameter
Suppose X1 , . . . , Xn is a random sample from a distribution with pdf
f (x; ), . Instead of listing all of the individual observations
X1 , . . . , Xn , we might prefer to give only the sample mean X o
Solution to Homework # 9, Stat 517
1. 7.2.1.
Solution: The likelihood function is
n
L() =
i=1
Therefore,
n
i=1
X2
1
1
1
i
e 2 = ( n/2 e 2
2
n
i=1
2
Xi
1
)( )n .
2
Xi2 is a sucient statistic for .
2. 7.2.4.
Solution: The likelihood function is
n
L() =
[(1
Solution to Homework # 9, Stat 517
1. 6.4.3.
Solution: Let = (1 , 2 ). The density is
f (x) =
1 x1
e 2 I(1 ,) (x).
2
The likelihood function is
n
L() =
f (Xi ) =
i=1
1 n(X1 )
2
e
I(1 ,) (X(1) ).
n
2
Thus, 1 = X(1) . To compute the MLE of 2 , we have
() =
S OME E LEMENTARY S TATISTICAL I NFERENCES
Chapter 5: Some Elementary Statistical Inferences
Sampling Statistics
Order Statistics
More on Condence Intervals
Introduction to Hypothesis Tests
S OME E LEMENTARY S TATISTICAL I NFERENCES
Sampling and Stati
M AXIMUM L IKELIHOOD M ETHODS
Rao-Cramer Lower Bound and Efciency
In this section, we give a lower bound on the variance of any unbiased
estimate.
We show that, under regularity conditions, the variances of the
maximum likelihood estimates achieve this
S UFFICIENCY
Measures of Quality of Estimators
Suppose f (x; ) for is the pdf (pmf) of a random variable X .
Consider a point estimator Yn = u(X1 , . . . , Xn ) of .
Consistency:
Yn p , i.e., Yn is close to for large sample size.
MLE is consistent under
STAT 517
Practice Midterm Fall 2012
Name:
Please return this page with your solution after exam.
1. A randomly selected sample of n = 12 students at a university is asked, How much
did you spend for textbooks this semester? The responses, in dollars, are
*BEGINNING OF EXAMINATION*
1.
For a 2-year select and ultimate mortality model, you are given:
(i)
q [ x ] +1 = 0.95 q x +1
(ii)
l76 = 98,153
(iii)
l77 = 96,124
Calculate l [ 75] +1.
(A)
96,150
(B)
96,780
(C)
97,420
(D)
98,050
(E)
98,690
Exam MLC: Spring
STAT 517
Practice Midterm Spring 2013
Name:
Please return this page with your solution after exam.
1. A randomly selected sample of n = 12 students at a university is asked, How much
did you spend for textbooks this semester? The responses, in dollars, ar
Chapter 6
General Formulas
Net Premium => Present Value of Premium = Present Value of Benefits
Gross Premium => Present Value of Premium = Present Value of Benefits + Present Value of Expenses
Variances
Premium Continuous; Benefits at Moment of Death Annu
Annuity Symbols
Type of Coverage
Payable Continuously
n| ax n Ex ax n ax ax:n
1 Ax
ax
d
1 Ax:n
ax:n
d
n| ax n Ex ax n ax ax:n
ax:n an n|ax
ax:n an n|ax
ax
whole life annuity
n-year temporary life annuity
n-year deferred life annuity
n-year certain and
Chapter 4
Life Insurance and Endowments
Type of
Coverage
Payable at Moment of Death
Payable at End of Year
Ax vt t px x t dt
whole life
n-year term
insurance
n-year pure
endowmen
t
n-year
endowmen
t insurance
u-year
deferred
n-year term
n-year
deferred
wh
U NBIASEDNESS, C ONSISTENCY, AND L IMITING D ISTRIBUTIONS
Chapter 4. Unbiasedness, Consistency, and Limiting Distributions
Expectations of Functions
Convergence in Probability
Convergence in Distribution
U NBIASEDNESS, C ONSISTENCY, AND L IMITING D ISTRIB