Question)1:

Express the following in summation notation:

a.  x1 + x2 + x3 + x4 + x5

b.  x1 + 2x2 + 3x3 + 4x4 + 5x5

c.   x1(2) + y1(2)+ x2(2) + y2(2)+ x 3(2) + y3(2) + … + x4(2) + y4(2)

Question 2:

Define the average (or mean) as x = 1 xi .  Show that:

a.  1 (xi  一 x) = 0                    

b.  1 (xi 一 x)xi  = 1 (xi 一 x)2

Question 3: (Wooldridge)Appendix)A)Q. 10):

Suppose that in a particular state a standardized test is given to all graduating high  school students. Let score denote a student’s score on the test. Someone discovers that performance on the test is related to the size of the student’s graduating high    school class.  The relationship is quadratic:

score = 46.5 + 0.082 * class 一 0.000147 class2

a.  How'do'you'literally'interpret'the'value'45.6'in'the'equation?''By'itself,'is'it'of much'interest?'Explain.

b.  From'the'equation,'what'is'the'optimal'size'of the'graduating'class'(i.e. the'class size'that'maximizes'the'test'score)?'(Round'your'answer'to'the'nearest'integer.) What'is'the'highest'achievable'test'score?

c.   Sketch'a'graph'that'illustrates'your'solutions'in'part'b.

d.  Does'it'seem'likely'that score and class have'a'deterministic'relationship?''That is,'is'it'realistic'to'think'that'once'you'know'the'size'of'a'student’s'graduating     class'you'know,'with'certainty,'his'or'her'test'score?''Explain.

Question 4 (Wooldridge)Appendix)B)Question)4):

For'a'randomly'selected'local'labour'market'area'in'Australia, let'X'represent'the     proportion'of'adults'over'age'65'who'are'employed,'i.e. the'mature'age'employment rate. The'X'is'restricted'to'a'value'between'zero'and'one.  Suppose'the'cumulative  distribution'function'for'X'is'given'by: F x  = 3x2 一 2x3 for'all 0 三 x 三 1. Find'the      probability'that'the'matureRage'employment'rate'is'at'least'0.6'(60%).

Question 5: (Wooldridge)Appendix C Question 1):

Let F", F$, F% and F& be'independent,'identically'distributed'random'variables'from'a   population'with'a'mean H and'variance I$ . Let F =  (F" + F$ + F% + F&) denote'the average of'these'four'random'variables.

a.  What'is'the'expected'value'and'variance'of F in'terms'of H and'variance I$ ?

b.  Now'consider'a'different'estimator'of H.

J =  F" +  F$ +  F% +  F&

This' is' an' example' of' a' weighted' average' of' the F0 . Show' that  W is' also' a weighted'average'of H. Find'the'variance'of W.

c.  Based'on'the'above,'which'estimator'of H do'you'prefer, F or W?

Question 6:

The following table gives the joint probability density function P(X = x,Y = y) = f (x, y)  of two random variables X and Y :

a.  Evaluate the marginal distributions of X and Y, fX(x) and fY(y).

b.  Evaluate E(X) and E( Y).

c.  Find the conditional distribution f (X = x|Y = 10) and its mean.

d.  Compare   E(X)   and   E(X| Y=10).   Are   the   conditional   and   unconditional expectations the same? If no, why are they different?

e.  Explain whether or not X'and Y are statistically'independent.

Question)7:

Let Xi  be the random variable which represents the return from a stock i.  There are 4 stocks with'the'mean and variance structure which'can'summarised as follows:

X1 ~ (1, 2),          X2 ~ (1, 2),          X3 ~ (2, 0),          X4 ~ (2, 4).

It is also known that