Q1.          Show that cov(aX,Y) = a cov(X,Y).

Ans:

E(X) is the expected value of the random variable X is the mean or average value of X.

COV(aX, Y)           = E([aX – E(aX)][Y – E(Y)])

                = E([aX – aE(X)][Y – E(Y)])

                                = E(a[X – E(X)][Y – E(Y)])

                                = aE([X – E(X)][Y – E(Y)])

                                = aCOV(X,Y)

2. State principle of least squares.

Ans:        Least squares, also called "regression analysis", is a computational procedure for fitting an equation to a set of experimental data points. The criterion of the "best" fit is that the sum of the squares of the differences between the observed data points, (Xi,Yi), and the value calculated by the fitting equation, Ycalc

(Xi), i.e.: SUM [Yi - Ycalc(Xi)]2 is a minimum. The reason for using the square is to make all of the differences positive numbers.

3. Find the coefficient of linear correlation between the variables X and Y given in the table.

X    1    3    4    6    8    9    11    14

Y    1    2    4    4    5    7    8      9

Also obtain the regression equation of Y on X for the data.

Ans:

Coefficient of Linear Correlation

Step-1: Calculate the average of X and Y

X_m = 56/8 = 7

Y_m = 40/5 = 8

Step-2: Calculate the standard deviation of X and Y data sets:

σ_x=√¹/_N((x₁-x_m)²+ …(x_n-x_m)²

=4.062

Similarly, σ_y=2.598

Step-3: Calculate the covariance between the two data sets.

σ_xy=¹/_N[(x₁-x_m)(y₁-y_m)+(x₂-x_m)(y₂-y_m)+ … +(x_n-x_m)(y_n-y_m)]

= 1/8[(1-7)(1-5) + (3-7)(2-5) + (4-7)(4-5) + (6-7)(4-5) + (8-7)(5-5) + (9-7)(7-5) + (11-7)(8-5)+(14-7)(9-5)]

= 10.5

Step-4: The correlation coefficient is defined as:

r =

= 10.5/(4.062 X 2.598) = 0.994970565 = 1 (approx).

Regression equation of y on x

Regression equation is y = a + bx

Where b=

 

a =         - b

Consider the following table:

xi             yi             xi2           xiyi

1              1              1              1

3              2              9              6

4              5              16            16

6              4              36            24

8              5              64            40

9              7              81            63

11            8              121          88

14            6              196          126

50            40            524          364

Last row is the sum of the each column.

n = 8, b = 8(364) – 56(40)/(8(524) – 56(56)) = 672/1056 = 0.636

a = 40/5 – 0.636(56/8) = 0.548

Thus, the equation is y = 0.548 + 0.636x.