Let the random variable X be representable as a sum of...

Let the random variable $X$ be representable as a sum of random variables $X=\sum_{i=1}^{n} X_{i} .$ Show that, if $\mathbf{E}\left[X_{i} X_{j}\right]=\mathbf{E}\left[X_{i}\right] \mathbf{E}\left[X_{j}\right]$ for every pair of $i$ and $j$ with $1 \leq i<j \leq n$, then $\operatorname{Var}[X]=\sum_{i=1}^{n} \operatorname{Var}\left[X_{i}\right]$.

Key Concepts

Variance of a Random Variable

Variance measures the spread of a random variable around its mean. It is defined as the expected value of the squared deviation from the mean, quantifying how much values of the variable differ from the mean on average. This concept is fundamental in probability and statistics as it provides insight into the uncertainty or variability present in the data.

Linearity of Expectation

The linearity of expectation is the property that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether the variables are independent. This property simplifies analysis in many situations, allowing us to compute expectations of complex expressions by breaking them down into simpler components.

Covariance

Covariance is a measure of how much two random variables change together. It is defined as the expected value of the product of their deviations from their respective means. A zero covariance indicates that there is no linear relationship between the variables, a condition that is crucial when simplifying the variance of a sum of random variables.

Sum of Random Variables and Variance Addition

When a random variable is represented as a sum of other random variables, the variance of the sum can be expressed in terms of the variances of the individual components and their covariances. Specifically, if the random variables are pairwise uncorrelated (i.e., their covariance is zero), the variance of the sum is simply the sum of their variances. This principle is key in many probabilistic analyses and applications.

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