91Ó°ÊÓ

Covariance measures the extent to which two random variables change together. Essentially, it tells us whether an increase in one variable will result in an increase, decrease, or no change in the other variable.
In a multinomial distribution, you can calculate the covariance between any two variables using the formula \[ \text{Cov}(X_i, X_j) = -np_i p_j \].
As seen in the exercise, for the variables \(X_1\) and \(X_2\) with parameters \(n = 3\) and probabilities \(p_1 = p_2 = p_3 = \frac{1}{3}\), we compute:

• \( \text{Cov}(X_1, X_2) = -3 \times \frac{1}{3} \times \frac{1}{3} = -\frac{1}{3} \).

The negative value indicates that these variables move in opposite directions, which is typical within a multinomial framework, where an increase in one outcome often causes a decrease in others due to the fixed total number of trials (n).

Correlation, denoted as \(\rho\) (rho), is a standardized measure used to determine the strength and direction of a linear relationship between two variables. Unlike covariance, which can take any value, correlation will always lie between -1 and 1.
A correlation of -1 means a perfect negative linear relationship, zero means no linear relationship, and 1 means a perfect positive linear relationship.
For our exercise, the correlation between \(X_1\) and \(X_2\) is calculated using:\[ \rho(X_1, X_2) = \frac{\text{Cov}(X_1, X_2)}{\sqrt{\text{Var}(X_1) \cdot \text{Var}(X_2)}} \]Given:

• \(\text{Cov}(X_1, X_2) = -\frac{1}{3}\)
• \(\text{Var}(X_1) = \text{Var}(X_2) = \frac{2}{3}\)

the correlation is:

• \(\rho(X_1, X_2) = \frac{-\frac{1}{3}}{\sqrt{\frac{2}{3} \times \frac{2}{3}}} = -\frac{1}{2}\).

This negative correlation value confirms that there's a consistent inverse relationship between the two, supporting the principle that an increase in one category's count may decrease another's in the multinomial distribution.

Variance provides a numerical value that indicates how spread out the values of a random variable are from its mean.
In essence, it measures the expected degree of deviation from the mean. For a multinomial distribution, the variance for a category \(X_i\) is given by:\[ \text{Var}(X_i) = np_i(1-p_i) \]Substituting our specific situation with \(n = 3\) and \(p_1 = p_2 = p_3 = \frac{1}{3}\), we find:

• \( \text{Var}(X_1) = 3 \times \frac{1}{3} \times \left(1 - \frac{1}{3}\right) = \frac{2}{3} \)

This calculation shows that for each category in this multinomial setup, the counts tend to vary around their expected values.
The variance gives insight into how much fluctuation is expected. However, it's essential to recognize that variance does not imply the direction of the variation like covariance and correlation do.