Given any two random variables X and Y, by the linearity of...

Given any two random variables $X$ and $Y$, by the linearity of expectations we have $\mathbf{E}[X-Y]=\mathbf{E}[X]-\mathbf{E}[Y]$. Prove that, when $X$ and $Y$ are independent, $\operatorname{Var}[X-Y]=\operatorname{Var}[X]+\operatorname{Var}[Y] .$

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Key Concepts

Linearity of Expectation

This principle asserts that the expected value of a linear combination of random variables is equal to the same linear combination of their expected values. It holds regardless of whether the random variables are dependent or independent, making it a powerful and widely applicable tool in probability theory.

Variance

Variance quantifies the spread or dispersion of a random variable's values around its mean. It is defined as the expected value of the squared deviation from the mean, and it provides insight into the reliability and variability of the outcomes.

Independence

Independence between random variables means that the occurrence or outcome of one variable does not affect the other. This property significantly simplifies analyses—particularly in variance calculations—since for independent variables, many cross terms (such as covariance) become zero.

Covariance

Covariance measures how two random variables co-vary or move together. When the random variables are independent, the covariance is zero, which is crucial in simplifying calculations like deriving the variance of the difference between two independent random variables.

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