The joint probability distribution is x -1 0 0 1 y 0 -1 1

step 1 Okay, so for exercise 43, we're first going to say that the correlation is zero. So we have the

The joint probability distribution is x -1 0 ...

The joint probability distribution is
$$
\begin{array}{lcccc}
x & -1 & 0 & 0 & 1 \\
y & 0 & -1 & 1 & 0 \\
f_{X Y}(x, y) & 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4
\end{array}
$$
Show that the correlation between $X$ and $Y$ is zero but $X$ and $Y$ are not independent.

Key Concepts

Independence

Two random variables are independent if the occurrence of one does not affect the probability distribution of the other. In terms of their joint probability distribution, independence means that the joint distribution can be expressed as the product of the marginal distributions. Lack of independence implies that knowing the value of one variable gives some information about the other.

Correlation

Correlation is a standardized measure of the linear relationship between two random variables. It is obtained by dividing the covariance by the product of the variables' standard deviations, resulting in a value between -1 and 1. A correlation of zero signifies no linear relationship, but, as with covariance, it does not guarantee independence between the variables.

Covariance

Covariance measures the degree to which two random variables vary together. It is calculated as the expected value of the product of the deviations of each variable from their individual means. A covariance of zero indicates a lack of linear relationship, although it does not imply that the variables are statistically independent.

Expected Value

The expected value, or mean, of a random variable is a fundamental measure that calculates the weighted average of all possible values of the variable, with the weights being their corresponding probabilities. Expected values are crucial for computing other statistical measures like variance and covariance.

Marginal Distribution

The marginal distribution of a random variable is derived from a joint probability distribution by summing (or integrating, in continuous cases) over the other variable(s). It reflects the probabilities of the variable in question without considering any association with other variables.

Joint Probability Distribution

A joint probability distribution gives the probabilities of different outcomes for two or more random variables taken together. It is essential for understanding how variables interact, as it allows us to compute probabilities, expectations, and other measures by considering the combination of values each random variable can take.

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