### QUESTIONS FROM THIS PAPER

mining – 这是利用mining进行训练的代写, 对mining的流程进行训练解析, 是有一定代表意义的mining等代写方向

### QUESTIONS FROM THIS PAPER

- State any theorems/conditions that you use in deter mining your solutions.

a) Commencing from first principles derive an expression for the Hessian (not the

```
Bordered hessian) for a function of two independent variable. You may assume
```

```
without proof that the two-point Taylor series of f(x,y) exists. [6 marks]
```

b) Commencing with the Cobb-Douglas function below:

#### (,)=

```
1 / 4
```

```
3 / 4
```

```
Optimize q subject to the constraint
```

#### 100 = 2 + 4

```
obtain this maximum using the method of Lagrange multipliers. [ 6 marks]
```

c) A student wishes to allocate her available study time of 60 hours per week between

```
two subjects in such a way as to maximize her grade average. She formulates two
```

```
functions governing the grades of each module which can be taken as:
```

```
1
```

#### (

```
1
```

#### )= 20 + 20

```
1
```

#### ,

```
1
```

#### > 0

```
and
```

```
2
```

#### (

```
2
```

#### )= 80 + 3

```
2
```

#### ,

```
2
```

#### > 80 / 3

```
maximize the grade average subject to the time constraint
```

```
1
```

#### +

```
2
```

```
= 60. You do not
```

```
need to show that you have obtained a maximum. [6 marks]
```

d) To be awarded marks this question must be done using Vector Calculus and the

```
Lagrange multiplier.
```

```
Optimize: the function
```

#### (

#### ,

#### )

#### = 49

```
2
```

```
2
```

```
subject to + 3 = 10.
```

```
Hint: You do NOT need to determine the Bordered Hessian for the critical
```

```
value, you simply have to locate it. [ 7 marks]
```

Question 2

Difference equations, Laplace transforms and the Calculus of Variations.

a) Suppose we have the following 1

```
st
```

```
order difference equation
```

```
```

#### = 1. 2

```
1
```

#### + 198 ,

find the particular solution using any method of your choice. Take 0

```
= 50. [5 marks]
```

```
b) Using only Laplace transforms solve the following Samuelson model given below i.e.
```

```
the second order difference equation (where
```

```
```

```
is national income):
```

```
+ 2
```

#### 5

```
+ 1
```

#### + 6

```
```

#### = 4

```
```

#### ,

```
```

#### = 0 < 0 ,

```
0
```

#### = 0 ,

```
1
```

#### = 1

```
You may use without proof that
```

```
1
```

#### [

```
1
```

```
```

```
( 1
```

```
```

```
)
```

#### ]=

#### (

#### )

#### =

```
```

#### <+ 1.

```
[ 10 marks]
```

```
c) Using the Euler-Lagrange equation of the Calculus of Variations
```

```
Optimize
```

#### (

#### 16

```
2
```

#### + 144 + 11

```
```

#### 4 ()

```
2
```

#### )

```
1
```

```
0
```

```
Subject to
```

#### (

#### 0

#### )

```
= 8 and
```

#### (

#### 1

#### )

#### = 8. 6 , 0 <

#### (

#### )

```
< 1 [ 10 marks]
```

```
3) State any theorems/conditions that you use in determining your solutions.
```

a) You buy a mining site, including exploration rights and there are set up costs of 285m

(million). You expect to extract the following value of gold over the next 6 years, net of

running costs: 40m, 73.5m, 123.5m, 90.5m, 54.5m and 21m. At the end of year 6 you

pay 30m clean-up costs. The site will then be handed back to authorities (as worthless). By

calculating the NPV (or otherwise) determine whether one should you go ahead with the

project? The cash flows are discounted at 6.8% p.a. [ 5 marks]

b) Explain what is meant by the internal rate of return (IRR) in the context of project

appraisal. What are the drawbacks of the IRR method? [ 5 marks]

```
i) Discuss the pros and cons of the various numerical methods such as the
```

```
bisection method, linear interpolation technique, the Newton-Raphson method
```

```
and the secant method in determining the IRR. You should also clearly discuss
```

```
any methods used in determining the initial iterate. [ 5 marks]
```

```
ii) Suppose one estimates that they can afford to repay 12 00 a month for 2 5
```

```
years on a mortgage. Interest is calculated at 4.3% p.a., payable monthly.
```

```
How large a mortgage can the individual afford? [5 marks]
```

```
iii) How much would an investor pay now (beginning of the month) for an
```

```
annuity, which pays 1, 5 00 at the end of each month for 10 years, if the
```

```
current interest rate is 12 % p.a. compounded weekly? [5 marks]
```

- State any theorems that you use in determining your solution.

a) Explain the meaning and use of each of the following terms:

```
i) A BLUE estimator [2 marks]
```

```
ii) A ( 1 )% confidence interval. [2 marks]
```

```
iii) Statistical significance. [2 marks]
```

```
vi) A one tailed test. [2 marks]
```

```
vii) A p-value. [2 marks]
```

```
viii) A biased estimator. [2 marks]
```

b) Suppose we want to compare the mean daily sales of two restaurants located in the

```
same town. The restaurants total sales for each of 12 randomly selected days are
```

```
given below:
```

```
Day
```

```
Restaurant1 (X
1
```

#### )

```
Restaurant
```

#### (X

```
2
```

#### )

```
Difference X
1 -
```

#### X

```
2
```

#### 1 1005 918 87

#### 2 2073 1971 102

#### 3 873 825 48

#### 4 1074 999 75

#### 5 1932 1827 105

#### 6 1338 1281 57

#### 7 1449 1302 147

#### 8 759 678 81

#### 9 1905 1782 123

#### 10 693 639 54

#### 11 2106 2049 57

#### 12 981 933 48

Conduct a paired t-test to determine whether there is a significant difference in the mean

sales, use 5% significance. [ 8 marks]

```
c) Show that the sample variance is an unbiased estimator of the population variance.
```

```
[ 5 marks]
```

- State any theorems that you use in determining your solution.

```
a) By using partial differentiation derive the equations that satisfy in
```

```
```

#### =+

```
```

#### +

```
```

```
where symbols have their usual meaning. [ 8 marks]
```

```
b) Suppose you are given the following information for an OLS regression:
```

```
```

```
```

#### = 830102 ,= 22 ,= 416. 5 ,= 86. 65 ,

```
```

```
2
```

#### = 3919654 ,= 130. 6

```
i) Calculate the ordinary least square regression line. [4 marks]
```

```
ii) Calculate () () [2 marks]
```

```
iii) Perform the hypothesis test such that
```

```
0
```

#### := 60

```
1
```

#### : 60

```
[2 marks]
```

```
iv) Perform the hypothesis test such that
```

```
0
```

#### := 0. 4

```
1
```

#### : 0. 4

```
[2 marks]
```

```
In both cases use a significance of 5%.
```

```
c) Discuss using appropriate mathematics the properties and assumptions underlying
```

```
the OLS technique, you should mention and discuss all 5 assumptions. [ 7 marks]
```

```
6) State any theorems that you use in determining your solution.
```

```
a) Suppose you are given model with two explanatory variables such that:
```

```
```

#### =+

```
1
```

```
1
```

#### +

```
2
```

```
2
```

#### +

```
```

#### , = 1 , 2 ,…

Using partial differentiation derive expressions for the expressions for the intercept and slope

coefficients for the model above. [ 10 marks]

```
b) A production function is specified as:
```

```
```

#### =+

```
1
```

```
1
```

#### +

```
2
```

```
2
```

#### +

```
```

#### , = 1 , 2 ,…,

```
```

#### ~( 0 ,

```
2
```

#### )

Where =log(), 1

```
=log( ),
```

```
2
```

```
=log ( )
```

The results are as follows:

```
1
```

#### = 10 ,

```
2
```

#### = 5 , = 12 ,

```
11
```

#### = 12 ,

```
12
```

#### = 8 ,

```
22
```

#### = 12 ,

```
1
```

#### = 10 ,

```
2
```

#### = 8 ,

```
```

#### = 10 ,

#### = 23 ( )

```
i) Compute estimates for the intercept, the slope coefficients and
```

```
2
```

. [5 marks]

```
ii) Show that (
```

```
1
```

#### )

```
= 0. 102. [4 marks]
```

```
iii) Test the hypotheses:
```

```
1
```

#### = 1

```
2
```

```
= 0 , separately at the 5% significance level.
```

```
You may take without calculation that
```

#### (

#### )

```
= 0. 78 and (
```

```
2
```

#### )

#### = 0. 102

```
[3 marks]
```

```
iv) Find a 95% confidence interval for the estimate
```

```
2
```

. [ 3 marks]

```
Variances and standard errors of Ordinary Least Squares estimators
```

#### (

```
1
```

#### )=

```
1
```

```
2
```

#### =

```
```

```
2
```

```
```

```
2
```

#### .

```
2
```

Note this formula involves both lower and upper case X

#### (

```
1
```

```
)=var(
```

```
1
```

#### )

#### (

```
2
```

#### )=

```
```

```
2
```

```
2
```

#### =

```
2
```

```
```

```
2
```

#### (

```
2
```

```
)=var(
```

```
2
```

#### )

#### 2

#### =

```
```

```
2
```

#### 2

```
```

```
2
```

#### =(

```
```

```
```

#### )

```
2
```

The coefficient of determination

```
2
```

#### = 1

```
```

```
2
```

```
```

```
2
```

```
Variances and standard errors of Multiple Ordinary Least Squares estimators (
```

```
explanatory variables)
```

#### (

```
1
```

#### )

#### =

```
2
```

```
11
```

#### ( 1

```
12
```

```
2
```

#### )

#### (

```
2
```

#### )=

```
2
```

```
22
```

#### ( 1

```
12
```

```
2
```

#### )

#### (

```
1
```

#### ,

```
2
```

#### )=

```
2
```

```
12
```

```
2
```

```
12
```

#### ( 1

```
12
```

```
2
```

#### )

#### ()=

```
2
```

#### +

```
1
```

```
2
```

#### (

```
1
```

#### )+ 2

```
1
```

```
2
```

#### (

```
1
```

#### ,

```
2
```

#### )+

```
2
```

```
2
```

#### (

```
2
```

#### )

#### (,

```
1
```

#### )=[

```
1
```

#### (

```
1
```

#### )+

```
2
```

#### (

```
1
```

#### ,

```
2
```

#### )]

#### (,

```
2
```

#### )=[

```
1
```

#### (

```
1
```

#### ,

```
2
```

#### )+

```
2
```

#### (

```
2
```

#### )]

#### =

```
```

```
1
```

```
1
```

```
2
```

```
2
```

```
2
```

#### =

```
1
```

```
1
```

#### +

```
2
```

```
2
```

```
```

#### CONTINUED

#### PLEASE TURN OVER

#### CONTINUED

#### PLEASE TURN OVER

#### CONTINUED

#### PLEASE TURN OVER

#### CONTINUED

#### PLEASE TURN OVER

#### CONTINUED

#### PLEASE TURN OVER

#### CONTINUED

#### END OF PAPER

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