Expected value:
E(X) = 0.42 + 0.21 + 0.07 = 0.7
E(Y) = 0.12 + 0.42 + 0.06 + 0.21 + 0.06 + 0.03 + 0.07 + 0.02 + 0.01 = 1.03
Variance:
Var(X) = (0.42 - 0.7)^2 + (0.21 - 0.7)^2 + (0.07 - 0.7)^2 = 0.21
Var(Y) = (0.12 - 1.03)^2 + (0.42 - 1.03)^2 + (0.06 - 1.03)^2 + (0.21 - 1.03)^2 + (0.06 - 1.03)^2 + (0.03 - 1.03)^2 + (0.07 - 1.03)^2 + (0.02 - 1.03)^2 + (0.01 - 1.03)^2 = 0.51
Standard deviation:
SD(X) = √0.21 = 0.46
SD(Y) = √0.51 = 0.71
Interpretation of asymmetry measure (α3):
The asymmetry measure (α3) is the third moment of the probability density function (PDF) of the compound random variables X and Y. The third moment (µ3) is equal to 0.0001, which indicates that the PDF is symmetric around the mean.
Interpretation of kurtosis measure (α4):
The kurtosis measure (α4) is the fourth moment of the probability density function (PDF) of the compound random variables X and Y. The fourth moment (µ4) is equal to 0.006, which indicates that the PDF is platykurtic, meaning that it has a lower peak than a normal distribution.
Covariance:
Cov(X,Y) = (0.42 - 0.7)(0.12 - 1.03) + (0.21 - 0.7)(0.42 - 1.03) + (0.07 - 0.7)(0.06 - 1.03) + (0.21 - 0.7)(0.21 - 1.03) + (0.07 - 0.7)(0.06 - 1.03) + (0.07 - 0.7)(0.03 - 1.03) + (0.21 - 0.7)(0.07 - 1.03) + (0.07 - 0.7)(0.02 - 1.03) + (0.07 - 0.7)(0.01 - 1.03) = -0.09
Pearson correlation coefficient:
ρxy = Cov(X,Y) / (SD(X) * SD(Y)) = -0.09 / (0.46 * 0.71) = -0.14
Conclusion:
The Pearson correlation coefficient (ρxy) is -0.14, which indicates that the variables X and Y are weakly negatively correlated. Therefore, the variables X and Y are not independent.
) The probability density function of the X and Y compound random variables is given below.
X Y 0 1 2
0 0.12
0.42
0.06
1 0.21
0.06
0.03
2 0.07 0.02 0.01
Accordingly, calculate the expected value, variance and standard deviation; Interpret the asymmetry measure (3) when the 3rd moment (µ3 = 0.0001) according to the arithmetic mean and the kurtosis measure (4) when the 4th moment (µ4 = 0.006) according to the arithmetic mean. Then, find the covariance (Cov (X, Y)) and the Pearson correlation coefficient (xy) to determine whether the variables X and Y are independent.
1 answer