Question 1:

Express the following in summation notation:

a.  x1 + x2 + x3 + x4 + x5

b.  x1 + 2x2 + 3x3 + 4x4 + 5x5

c.  ox1(2) + y1(2)) +0x2(2) + y2(2) +0x3(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:

scOTe = 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: FOx = 3x2  一 2x3  for all 0 ≤ x ≤ 1. Find the

probability that the mature-age employment rate is at least 0.6 (60%).

Question 5: (Wooldridge Appendix C Question 1):

Let Y1, Y2, Y3  and Y4 be independent, identically distributed random variables from a    population with a mean μ and variance σ2 . Let Y = Y1  + Y2  + Y3  + Y4) denote the average of these four random variables.

a.  What is the expected value and variance of Y in terms of μ and variance σ2 ?

b.  Now consider a different estimator of μ .                  

1            1           1           1

w =   Y1 +   Y2  +  Y3  +  Y4

This  is  an  example  of a weighted  average  of the  Yi .  Show  that  W is  also  a weighted average of  μ .  Find the variance of W.

c.  Based on the above, which estimator of μ do you prefer, Y 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