Suppose X and Y have the joint discrete distribution shown...

Suppose $X$ and $Y$ have the joint discrete distribution shown in the table. Show that $X$ and $Y$ are uncorrelated but not independent. $$ \begin{array}{|cc|ccc|} \hline X & Y & -1 & 0 & 1 \\ \hline-1 & & 0 & \frac{1}{4} & 0 \\ 0 & & \frac{1}{4} & 0 & \frac{1}{4} \\ 1 & & 0 & \frac{1}{4} & 0 \\ \hline \end{array} $$

Suppose $X$ and $Y$ have the joint discrete distribution shown in the table. Show that $X$ and $Y$ are uncorrelated but not independent.
$$
\begin{array}{|cc|ccc|}
\hline X & Y & -1 & 0 & 1 \\
\hline-1 & & 0 & \frac{1}{4} & 0 \\
0 & & \frac{1}{4} & 0 & \frac{1}{4} \\
1 & & 0 & \frac{1}{4} & 0 \\
\hline
\end{array}
$$

Key Concepts

Uncorrelated Random Variables

When random variables are uncorrelated, their covariance is zero, implying no linear association between them. However, this does not necessarily mean that the variables are independent, as they might still have a nonlinear relationship.

Independence of Random Variables

Independence between random variables means that the occurrence of one variable does not affect the distribution of the other. This is mathematically expressed by the joint distribution being equal to the product of the marginal distributions, implying no form of dependency between the variables.

Covariance

Covariance measures the degree to which two random variables change together. A covariance of zero indicates that there is no linear relationship between the variables, although other types of dependencies may still be present.

Expected Value

The expected value is the measure of the central tendency of a random variable, calculated as the weighted average of all possible values, with weights being their corresponding probabilities. It serves as the foundation for further statistical measures such as variance and covariance.

Marginal Distribution

The marginal distribution is obtained by summing or integrating the joint distribution over one or more variables, thus providing the probability distribution for a single variable irrespective of the others. This is crucial for computing expectations and variances for individual variables.

Joint Distribution

The joint distribution describes the probability of all possible combinations of values for two or more random variables. It provides a comprehensive view of how the variables interact and is essential for analyzing their combined behavior.

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