Download Fall 2008

The written Master’s Examination
Option Probability and Statistics
Fall 2008
Full points may be obtained for correct answers to 8 questions. Each numbered question
(which may have several parts) is worth the same number of points. All answers will be
graded, but the score for the examination will be the sum of the scores of your best 8
solutions.
Use separate answer sheets for each question. DO NOT PUT YOUR NAME ON
YOUR ANSWER SHEETS. When you have finished, insert all your answer sheets
into the envelope provided, then seal and print your name on it.
Any student whose answers need clarification may be required to submit to an oral
examination.
MS Exam, Option Probability and Statistics, FALL 2008
1. (Stat 401)
Suppose the random variables X 1 and X 2 have the joint probability mass function
x
⎛ x ⎞⎛ 1 ⎞ 1 ⎛ x ⎞
p(x1 , x 2 ) = ⎜ 1 ⎟ ⎜ ⎟ ⎜ 1 ⎟ , x2 = 0,1,..., x1 , x1 = 1,2 ,
⎝ x2 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠
zero elsewhere. Determine:
(a) E ( X 2 ) ;
(b) u(x1 ) = E(X 2 | X1 = x1 ) ;
(c) E[u ( X 1 )] .
2. (Stat 411)
Let X1 ,..., X n be a random sample from a uniform distribution with density function
1
f ( x, θ ) = , 0 < x ≤ θ,
θ
and parameter θ > 0 .
(a) Find the MLE θ̂ of the parameter θ .
(b) Is θ̂ a minimal sufficient statistic? Why?
(c) Find an MVUE for θ .
3. (Stat 411)
A random sample X1 ,..., X n arises from a continuous distribution with density f, given by
H 0 : f (x; θ) = 1/ θ , 0 < x ≤ θ , zero elsewhere,
or
H1 : f (x; θ) = (1/ θ) e − x / θ , 0 < x < ∞ , zero elsewhere.
(a) Find the likelihood ratio test statistic associated with the test of H 0 against H1 ;
(b) Show that the critical region of the associated likelihood ratio test is
n
{(x1 ,..., x n ) | (1/ n)∑ x i ≤ c max{x i }}
i =1
for some constant c > 0 .
2
1≤i ≤ n
MS Exam, Option Probability and Statistics, FALL 2008
4. (Stat 416)
Diet A was given to a group of 10 overweight boys between the ages of 8 and 10. Diet B was given to
another independent group of 10 similar overweight boys. The weight loss in pounds is given below.
Diet A:
7.2
2.1
3.3
0.1
4.7
6.2
0.3
1.5
4.2
6.1
Diet B:
5.3
6.6
4.9
7.3
8.4
9.1
7.6
2.4
3.5
4.3
To test equal effects of the diets against the alternative that diet B is more effective, make appropriate
model assumptions, set up the null hypothesis and the alternative in terms of the parameter in the
model, and then use Wilcoxon’s Rank Sum Test at the level of α = 0.05 . Utilize that under the null
hypothesis the distribution of the sum of ranks for either diet has a 0.947 quantile of 127 and a 0.955
quantile of 128.
5. (Stat 431)
A client has a finite population of 7 units and has funds to survey 3 units for the purpose of estimating
the mean and its related standard deviation. There are two sampling plans to be considered.
Sampling Plan 1: A simple random sample of size 3 without replacement, SRS (7, 3),
Sampling Plan 2: A uniform sampling plan on the following support.
Samples in the support: {1,2,3 }, {3,4,5}, {1,5,6}, {1,4,7}, {2,4,6}.
Suppose upon the survey we obtain the following data: 27, 30, and 33.
1- What are the HT estimator and its value of the population mean under both sampling plans?
2- What is the standard deviation of your estimator under both sampling plans?
3- Which sampling plan would you recommend to your client and why?
6. (Stat 461)
Let {N(t), t ≥ 0} be a Poisson process with rate λ , that is independent of the nonnegative random
variable T with mean μ and variance σ2 . Determine
(a)
Cov(N(T), T) ,
(b)
Var(N(T)) .
3
MS Exam, Option Probability and Statistics, FALL 2008
7. (Stat 471)
Use the Lindo output, given on the next page, to answer the queries below to the following problem.
A snack mix company mixes cashews, peanuts, and pretzels in different proportions to make three
types of snack packs that are sold for three different prices:
Cashews Peanuts Pretzels
Exotic ⎡
Super ⎢⎢
Tasty ⎢⎣
3
2
1
3
1
2
1
2
3
⎤
⎥
⎥
⎥⎦
All measures are in ounces. A packet of Exotic sells for $ 3.00, Super sells for $ 2.50, and Tasty sells
for $ 1.50.
The available supply: 100 Oz of cashews, 150 Oz of peanuts, and 200 Oz of pretzels.
1. If the company has to pay for cashews an additional 30 cents per Oz, what will be the gross profit?
2. If the supply of cashews were 120 Oz, what would be the gross profit?
3. If pretzels had a price reduction of $ .05 per Oz, what would be the gain in gross profit?
4
MS Exam, Option Probability and Statistics, FALL 2008
Lindo output:
MAX 3 EXOTIC +2.5 SUPER + 1.5 TASTY
SUBJECT TO
3 EXOTIC + 2 SUPER + TASTY <= 100
EXOTIC + 3 SUPER + 2 TASTY <= 150
EXOTIC + 2 SUPER + 3 TASTY <= 200
END
**************************************************************
LP OPTIMUM FOUND AT STEP 3
OBJECTIVE FUNCTION VALUE
1)
133.3333
VARIABLE
VALUE
EXOTIC
8.333333
SUPER
8.333333
TASTY
58.333332
ROW
2)
3)
4)
REDUCED COST
0.000000
0.000000
0.000000
SLACK OR SURPLUS DUAL PRICES
0.000000
0.916667
0.000000
0.166667
0.000000
0.083333
NO. ITERATIONS=
3
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE
CURRENT ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
EXOTIC
3.000000
0.500000
1.000000
SUPER
2.500000
0.200000
0.250000
TASTY
1.500000
0.500000
0.142857
RIGHTHAND SIDE RANGES
ROW
CURRENT
ALLOWABLE
ALLOWABLE
RHS
INCREASE
DECREASE
2 100.000000
99.999992
20.000000
3 150.000000
24.999998
12.499999
4 200.000000
20.000000
99.999992
5
MS Exam, Option Probability and Statistics, FALL 2008
8. (Stat 473)
Bob sells balloons at the entrance gate and Joe sells them at the exit gate of a zoo. The two gates are
one mile away from each other. Everyone who enters through the east gate exits through the west gate.
Every day there are exactly 1000 kids visiting the zoo. Parents hesitate initially to spend money and are
willing to buy at most one balloon per kid. At most 400 parents agree to consider buying at the
entrance gate, provided the price is reasonable. Initial demand = 400-price in cents at the entrance gate.
The kids who did not get any balloon clamor and cry and the parents promise to buy provided the price
is right. The demand at the exit gate = the number of children without balloon-price in cents. If Bob
and Joe know this, what will be their best strategy and what will be their optimal income?
9. (Stat 481)
A factorial experiment with 8 runs, as given below, is conducted with 4 factors at 2 levels. Let these
factor be denoted by A, B, C, and D. The factor levels for A, B, C, and D are coded by 1 and 0. Two
observations are collected under each run.
1- Write down a homogeneous linear fixed additive model for the responses when the parameters
in the model consist of the general mean, main effects, AB and AC interactions only.
2- Indicate how would you estimate unbiasedly the effects and the common variance σ2 associated
with the noise in the model using the notation you proposed in 1?
Run Number A
B
C
D
Observations
1
2
3
4
5
6
7
8
0
0
1
1
0
0
1
1
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
27, 30
26, 29
25, 26
38, 40
30, 31
25, 27
23, 25
24, 26
0
0
0
0
1
1
1
1
3- Explain how you would carry out the test at 5% significance level that factors A and C do not
interact in this experiment.
6
MS Exam, Option Probability and Statistics, FALL 2008
10. (Stat 481)
Consider one explanatory variable x and the response variable Y, where the model is
Yi = β0 + β1x i + εi , i = 1,..., n ,
and where the errors ε1 ,..., ε n are i.i.d. normally distributed with mean 0 and variance σ 2 .
(a) Derive the expression of least square estimates β̂0 and β̂1 of coefficients β0 and β1 , respectively,
such that the fitted line βˆ + βˆ x is the best straight line to fit the observed data (x , y ),..., (x , y ) ,
0
1
1
1
n
n
which is
n
βˆ 1 =
∑(x
i =1
n
∑(x
i =1
− x ) yi
i
i
− x)
2
, βˆ 0 = y − βˆ 1x .
(b) Prove that the sampling distribution for the least square estimator β̂1 is as follows:
−1
⎛
⎞
⎡ n
2⎤
βˆ 1 ~ N ⎜ β1 , σ 2 ⋅ ⎢ ∑ ( x i − x ) ⎥ ⎟
⎜
⎣ i =1
⎦ ⎟⎠
⎝
(c) When the error variance σ 2 is unknown, it could be replaced by the mean square error MSE. Let
( )
⎡ n
2⎤
s βˆ 1 = MSE ⋅ ⎢∑ ( x i − x ) ⎥
⎣ i =1
⎦
and show that
βˆ 1 − β1
~ t (n − 2) .
s βˆ
( )
1
[Note: Use Cochran’s Theorem.]
7
−1