) 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.
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.
To calculate the expected value (mean) of X and Y, we need to multiply each value by its corresponding probability and sum them up.
Expected value (mean) of X:
E(X) = (0 * 0.12) + (1 * 0.42) + (2 * 0.06) + (0 * 0.21) + (1 * 0.06) + (2 * 0.03) + (0 * 0.07) + (1 * 0.02) + (2 * 0.01)
= 0 + 0.42 + 0.12 + 0 + 0.06 + 0.06 + 0 + 0.02 + 0.02
= 0.7
Expected value (mean) of Y:
E(Y) = (0 * 0.12) + (1 * 0.42) + (2 * 0.06) + (0 * 0.21) + (1 * 0.06) + (2 * 0.03) + (0 * 0.07) + (1 * 0.02) + (2 * 0.01)
= 0 + 0.12 + 0.12 + 0 + 0 + 0.06 + 0 + 0.02 + 0.02
= 0.34
To calculate the variance of X and Y, we need to subtract the mean from each value, square it, multiply by its corresponding probability, and sum them up.
Variance of X:
Var(X) = [(0 - 0.7)^2 * 0.12] + [(1 - 0.7)^2 * 0.42] + [(2 - 0.7)^2 * 0.06] + [(0 - 0.7)^2 * 0.21] + [(1 - 0.7)^2 * 0.06] + [(2 - 0.7)^2 * 0.03] + [(0 - 0.7)^2 * 0.07] + [(1 - 0.7)^2 * 0.02] + [(2 - 0.7)^2 * 0.01]
= [(-0.7)^2 * 0.12] + [(0.3)^2 * 0.42] + [(1.3)^2 * 0.06] + [(-0.7)^2 * 0.21] + [(0.3)^2 * 0.06] + [(1.3)^2 * 0.03] + [(-0.7)^2 * 0.07] + [(0.3)^2 * 0.02] + [(1.3)^2 * 0.01]
= 0.0882 + 0.0378 + 0.078 + 0.0969 + 0.009 + 0.0513 + 0.0343 + 0.0018 + 0.0017
= 0.399
Variance of Y:
Var(Y) = [(0 - 0.34)^2 * 0.12] + [(1 - 0.34)^2 * 0.42] + [(2 - 0.34)^2 * 0.06] + [(0 - 0.34)^2 * 0.21] + [(1 - 0.34)^2 * 0.06] + [(2 - 0.34)^2 * 0.03] + [(0 - 0.34)^2 * 0.07] + [(1 - 0.34)^2 * 0.02] + [(2 - 0.34)^2 * 0.01]
= [(-0.34)^2 * 0.12] + [(0.66)^2 * 0.42] + [(1.66)^2 * 0.06] + [(-0.34)^2 * 0.21] + [(0.66)^2 * 0.06] + [(1.66)^2 * 0.03] + [(-0.34)^2 * 0.07] + [(0.66)^2 * 0.02] + [(1.66)^2 * 0.01]
= 0.1632 + 0.1782 + 0.7182 + 0.1632 + 0.0396 + 0.1098 + 0.0406 + 0.0211 + 0.007
= 1.4409
To calculate the standard deviation, we take the square root of the variance.
Standard deviation of X:
σX = √Var(X) = √0.399 ≈ 0.632
Standard deviation of Y:
σY = √Var(Y) = √1.4409 ≈ 1.200
The asymmetry measure (α3) is the standardized measure of skewness. Skewness measures the asymmetry of a probability distribution. A skewness measure of 0 indicates a perfectly symmetrical distribution. Positive skewness indicates a right-skewed distribution, where the tail is longer on the right side, and negative skewness indicates a left-skewed distribution, where the tail is longer on the left side.
Given µ3 = 0.0001, we can calculate α3 using the formula:
α3 = µ3 / σ^3
Plugging in the values:
α3 = 0.0001 / (0.632)^3 ≈ 0.00025
The kurtosis measure (α4) is the standardized measure of kurtosis. Kurtosis measures the "tailedness" of a probability distribution. A kurtosis measure of 0 indicates a Gaussian distribution. Positive kurtosis indicates heavy tails, and negative kurtosis indicates light tails.
Given µ4 = 0.006, we can calculate α4 using the formula:
α4 = µ4 / σ^4
Plugging in the values:
α4 = 0.006 / (0.632)^4 ≈ 0.0189
To determine whether the variables X and Y are independent, we can calculate the covariance (Cov(X, Y)) and the Pearson correlation coefficient (ρxy). If Cov(X, Y) = 0 and ρxy = 0, then X and Y are independent.
The covariance (Cov(X, Y)) is calculated as:
Cov(X, Y) = E(XY) - E(X)E(Y)
To calculate E(XY), we multiply the product of X and Y by their corresponding probability and sum them up:
E(XY) = (0*0*0.12) + (0*1*0.42) + (0*2*0.06) + (1*0*0.21) + (1*1*0.06) + (1*2*0.03) + (2*0*0.07) + (2*1*0.02) + (2*2*0.01)
= 0 + 0 + 0 + 0 + 0.06 + 0.06 + 0 + 0.02 + 0.04
= 0.18
Now we can calculate Cov(X, Y):
Cov(X, Y) = 0.18 - (0.7 * 0.34)
= 0.18 - 0.238
= -0.058
The Pearson correlation coefficient (ρxy) is calculated as:
ρxy = Cov(X, Y) / (σX * σY)
Plugging in the values:
ρxy = -0.058 / (0.632 * 1.200)
≈ -0.0808
Since Cov(X, Y) is not equal to 0 and ρxy is not equal to 0, X and Y are not independent.