Step 1: Find the mean of variable X. Sum up all the observations in variable X and divide the sum obtained with the number of terms. Thus, (80 + 63 + 100)/3 = 81.
Step 2: Subtract the mean from all observations. (80 - 81), (63 - 81), (100 - 81).
Step 3: Take the squares of the differences obtained above and then add them up. Thus, (80 - 81)2 + (63 - 81)2 + (100 - 81)2.
Step 4: Find the variance of X by dividing the value obtained in Step 3 by 1 less than the total number of observations. var(X) = [(80 - 81)2 + (63 - 81)2 + (100 - 81)2] / (3 - 1) = 343.
Step 5: Similarly, repeat steps 1 to 4 to calculate the variance of Y. Var(Y) = 633.333
Step 6: Choose a pair of variables.
Step 7: Subtract the mean of the first variable (X) from all observations; (80 - 81), (63 - 81), (100 - 81).
Step 8: Repeat the same for variable Y; (70 - 47), (20 - 47), (50 - 47).
Step 9: Multiply the corresponding terms: (80 - 81)(70 - 47), (63 - 81)(20 - 47), (100 - 81)(50 - 47).
Step 10: Find the covariance by adding these values and dividing them by (n - 1). Cov(X, Y) = [(80 - 81)(70 - 47) + (63 - 81)(20 - 47) + (100 - 81)(50 - 47)]/(3-1) = 260.
Step 11: Use the general formula for the covariance matrix to arrange the terms. The matrix becomes: \begin{bmatrix} 343 & 260\\ 260& 633.333 \end{bmatrix}
Sample covariance matrix is given by \frac{\sum_{1}^{n}\left ( x_{i} -\overline{x}\right )^{2} }{n-1} .
Here, μx = 84, n = 3
var(x) = [(92 - 84)2 + (60 - 84)2 + (100 - 84)2] / (3 - 1) = 448
Also, μy = 60, n = 3
var(y) = [(80 - 60)2 + (30 - 60)2 + (70 - 60)2] / (3 - 1) = 700
Now, cov(x, y) = cov(y, x) = [(92 - 84)(80 - 60) + (60 - 84)(30 - 60) + (100 - 84)(70 - 60)] / (3 - 1) = 520.
The population covariance matrix is given as: \begin{bmatrix} 448 & 520\\ 520& 700 \end{bmatrix}
Population variance is given by \frac{\sum_{1}^{n}\left ( x_{i} -\mu\right )^{2} }{n} .
Here, μx = 56.5, n = 4
var(x) = [(68 - 56.5)2 + (60 - 56.5)2 + (58 - 56.5)2 + (40 - 56.5)2 ] / 4 = 104.75
Also, μy = 30, n = 4
var(y) = [(29 - 30)2 + (26 - 30)2 + (30 - 30)2 + (35 - 30)2] / 4 = 10. 5
Now, cov(x, y) = \frac{\sum_{1}^{4}\left ( x_{i} -\mu_{x}\right )\left ( y_{i}-\mu_{y} \right ) }{4}
cov(x, y) = -27
The population covariance matrix is given as: \begin{bmatrix} 104.7 &-27 \\ -27& 10.5 \end{bmatrix}
Sample covariance matrix is given by \frac{\sum_{1}^{n}\left ( x_{i} -\overline{x}\right )^{2} }{n-1} .
n = 5,
- μx = 22.4, var(X) = 321.2 / (5 - 1) = 80.3
- μy = 12.58, var(Y) = 132.148 / 4 = 33.037
- μz = 64, var(Z) = 570 / 4 = 142.5
Now, cov(X, Y) = \frac{\sum_{1}^{5}\left ( x_{i} -22.4\right )\left ( y_{i}-12.58\right ) }{5-1} = -11.76
⇒ cov(X, Z) = \frac{\sum_{1}^{5}\left ( x_{i} -22.4\right )\left ( z_{i}-64 \right ) }{5-1} = 34.97
⇒ cov(Y, Z) = \frac{\sum_{1}^{5}\left ( y_{i} -12.58\right )\left ( z_{i}-64 \right ) }{5-1} = -40.87
The covariance matrix is given as:
\begin{bmatrix} 80.3 & -13.865 &14.25 \\ -13.865 & 33.037 & -39.5250\\ 14.25 & -39.5250 & 142.5 \end{bmatrix}