Let Y1 and Y2 be uncorrelated random variables and consider...

Let $Y_{1}$ and $Y_{2}$ be uncorrelated random variables and consider $U_{1}=Y_{1}+Y_{2}$ and $U_{2}=Y_{1}-Y_{2}$. a. Find the $\operatorname{Cov}\left(U_{1}, U_{2}\right)$ in terms of the variances of $Y_{1}$ and $Y_{2}$. b. Find an expression for the coefficient of correlation between $U_{1}$ and $U_{2}$. c. Is it possible that $\operatorname{Cov}\left(U_{1}, U_{2}\right)=0 ?$ When does this occur?

Let $Y_{1}$ and $Y_{2}$ be uncorrelated random variables and consider $U_{1}=Y_{1}+Y_{2}$ and $U_{2}=Y_{1}-Y_{2}$.
a. Find the $\operatorname{Cov}\left(U_{1}, U_{2}\right)$ in terms of the variances of $Y_{1}$ and $Y_{2}$.
b. Find an expression for the coefficient of correlation between $U_{1}$ and $U_{2}$.
c. Is it possible that $\operatorname{Cov}\left(U_{1}, U_{2}\right)=0 ?$ When does this occur?

Key Concepts

Covariance

Covariance measures how two random variables change together. When linear combinations of random variables are considered, the properties of covariance allow us to expand and express the covariance of sums or differences in terms of the covariances and variances of the individual random variables. Understanding these properties is essential when analyzing how transformations affect the relationship between variables.

Linear Transformations of Random Variables

Linear transformations involve creating new random variables by forming linear combinations of existing ones. This is important because it allows us to deduce the distributional properties of the transformed variables—including their covariance and variance—using the known properties of the original variables. Analyzing these transformations can reveal under what conditions the new variables might be uncorrelated (i.e., have zero covariance).

Variance

Variance is a measure of the spread or dispersion of a random variable around its mean. In the context of linear combinations, the variances of the underlying random variables are used to compute the variance of the new variables, as well as to normalize measures like the correlation coefficient. Recognizing how variance behaves under addition or subtraction is crucial for establishing relationships between transformed variables.

Correlation Coefficient

The correlation coefficient standardizes the covariance by the product of the variables' standard deviations, yielding a dimensionless measure that indicates the strength and direction of a linear relationship between two random variables. This coefficient is important because it provides a normalized scale, typically ranging from -1 to 1, which facilitates comparison across different studies or contexts even when the variables have different units or scales.

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