Let the discrete random variables Y1 and Y2 have the joint...

Let the discrete random variables $Y_{1}$ and $Y_{2}$ have the joint probability function $$p\left(y_{1}, y_{2}\right)=1 / 3, \quad \text { for }\left(y_{1}, y_{2}\right)=(-1,0),(0,1),(1,0)$$ Find $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)$. Notice that $Y_{1}$ and $Y_{2}$ are dependent. (Why?) This is another example of uncorrelated random variables that are not independent.

Let the discrete random variables $Y_{1}$ and $Y_{2}$ have the joint probability function
$$p\left(y_{1}, y_{2}\right)=1 / 3, \quad \text { for }\left(y_{1}, y_{2}\right)=(-1,0),(0,1),(1,0)$$
Find $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)$. Notice that $Y_{1}$ and $Y_{2}$ are dependent. (Why?) This is another example of uncorrelated random variables that are not independent.

Key Concepts

Expected Value

The expected value of a random variable is the long-run average value of the variable over numerous independent repetitions of the same experiment. In the context of joint distributions, expected values calculated over one variable or products of variables are essential in determining measures such as covariance.

Dependence versus Independence

Two random variables are independent if the occurrence of one does not affect the probability distribution of the other; i.e., their joint probability distribution factors into the product of their marginal distributions. When they are not independent, there is some form of dependence or relationship even if the linear correlation (covariance) between them is zero.

Covariance

Covariance is a statistical measure that describes the extent to which two random variables change together. It is calculated as the expected value of the product of the deviations of each variable from their means. A zero covariance indicates that there is no linear relationship between the variables, even though they may still be dependent in other ways.

Joint Probability Distribution

The joint probability distribution of discrete random variables is a function that gives the probability that the variables simultaneously take on a specific set of values. This function allows us to compute the probabilities of various events involving both variables and is central to the study of models involving multiple random variables.

Uncorrelated

Uncorrelated random variables are those that have a covariance of zero. It is important to understand that uncorrelatedness only implies the absence of a linear relationship, and does not necessarily mean that the variables are independent; there can still be other types of dependence present.

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