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18

2k. This statistic can be used for an overall rejection or acceptance of the null hypothesis based on the independent tests. [See G. S. Maddala, Econometrics (New York: McGraw-Hill, 1977), p. 48. The test is from C. R. Rao, Advanced Statistical Methods in Biometric Research (New York: Wiley, 1952), p. 44.] 32. The weekly cash inflows (x) and outflows (y) of a business firm are random variables. The following data give values of x and for 30 weeks. Assume that X and are normally distributed.

(a) Obtain unbiased estimates of the means and variances of x, y, and x - y. Also obtain 95% confidence intervals for these six variables.

(b) Test the hypothesis > p, assuming that:

(1) x and are independent.

(2) x and are correlated.

Appendix to Chapter 2

Matrix Algebra

In this appendix we present an introduction to matrix notation. This will facilitate the exposition of some material in later chapters-in particular. Chapter 4.

A matrix is a rectangular array of elements; for example,

"2 17 4" 12-21 4 1 2 -3

We shall denote this by A and say that it is of order 3x4. The first number is the number of rows, and the second number is the number of columns. A matrix of order 1 X n is called a row vector, and a matrix of m x 1 is called a

column vector; for example, b = (1, 2, 7, 3) is a row vector and =

8 -10



a column vector. Henceforth we shall follow the convention that of writing all column vectors without a prime and writing row vectors with a prime; for ex-

ample, if b =

is a column vector, b = [1, 2, 7, 3] is a row vector.

A transpose of a matrix. A, denoted by A, is the same matrix A with rows and columns interchanged. In the example above.

We shall now define matrix addition, subtraction, and multiplication. Matrix addition (or subtraction) is done by adding (subtracting) the corresponding elements and is defined only if the matrices are of the same order. If they are not of the same order, then there are no corresponding elements. For example, if

then

A + =

1 2 7 10 6

4 6 9 -4 -7 16

and =

2 10

A - =

-2 -2 5 2 7-4

Obviously, A + B is not defined because A is of order 2x3 and B is of order 3x2. Also note that A + = + A. As we shall see later, this commutative law does not hold for matrix multiplication; that is, AB and A need not be defined, and they need not be equal even if they are defined.

Multiplication of a Matrix by a Scalar

This is done by multiplying each element of the matrix by the scalar. For instance, if

2 3 1 0 1 7

then 4A =

8 12 4 0 4 28

Matrix Equality

Two matrices A and are said to be equal if they are of the same order and they have all the corresponding elements equal. In this case, A - = 0 (a matrix with all elements zero; such a matrix is known as a null matrix).

Scalar Product of Vectors

If b and are two vectors of the same order, so that b = (6,, bj, . . . , b„) and c = (c„ Cj, . . . , c„), the scalar product of the two vectors is defined as bc = fejCi + + • • • + b„c„. The multiplication is row-column wise and is



achieved by multiplying the corresponding elements and adding up the result. For example, if b = (2, - I, 2) and c = (0, 3, 3) then bc = [(2)(0) + (- 1)(3) + (2)(3)] = 0- 3 + 6 = 3. The scalar product is not defined if the vectors are not of the same order.

Matrix Multiplication

A matrix can be considered as a series of row vectors or a series of column vectors. Matrix multiplication is also done row-column wise. If we have two matrices and and we need the product EC, we write as a series of row vectors and as a series of column vectors and then take scalar products of the row vectors in and the column vectors in C. Clearly, for this to be possible the number of elements in each of the rows of should be equal to each of the columns of C. If is of order m x n and is of order n x k, is defined because the number of elements in the row vectors in and the column vectors in are both n. But if is n x k and is m x n, is not defined. If is an m X n matrix and is an x matrix, we can write as a set of m row vectors and as a set of column vectors; each of order n. That is.

Then EC is defined as

= [ ,, Cj, . . . , ]

b\C2

=

,

blc,

EC is of order m x k.li has as many rows as E and as many columns as C. As an example, consider

2 1 0 1

E is of order 2 x 3 and of order 2x2. Hence EC is not defined. But BC is defined.

EC =

2 0 1 1

(2)(7) + (0)(8)

(2)(5) + (0)(6)

10"

(1)(7) + (1X8)

(1)(5) + (1)(6)

(3)(7) + (2)(8)

(3)(5) + (2)(6)

Note that EC is not defined but CE is defined.

CB =

"7 5"

"2 1 3"

8 6

0 1 2

14 16

12 14

Note: Given two matrices, and C, one or both of the products EC and CE may not be defined, and even if they are both defined, they may not be of the same order, and even if they are of the same order, they may not be equal. For



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