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Understanding independent random variables is crucial for grasping concepts of variance in linear combinations. Independent random variables are those variables whose outcomes do not influence each other. For example, rolling two separate dice can be considered as having independent variables, as the result of one roll doesn't affect the other. In statistical terms, if you have a set of variables \(X_1, X_2, \ldots, X_p\), they are independent if the joint probability of the outcome can be expressed as the product of their individual probabilities. This mathematical property simplifies calculations, especially when you're dealing with variances because it allows easy summation without considering covariances. In our exercise, where we deal with a linear combination of independent variables, understanding their independence is key to effectively calculating the total variance. The independence allows for the variance of the sum to be calculated as the sum of individual variances, because no variable influences any other in this context.

Variance is a statistical measure that tells us how much the values of a random variable tend to vary. It's often denoted by \(V(X)\), where \(X\) can be any random variable. The variance quantifies the expectation of the squared deviation of a random variable from its mean, providing insights into the data's dispersion. There are key properties of variance that play a significant role when dealing with linear combinations:

• Variance of a Constant: The variance of a constant is zero since a constant value does not change.
• Variance of a Scaled Variable: If you multiply a random variable by a constant, the variance is scaled by the square of that constant, i.e., \( V(cX) = c^2 V(X) \).
• Sum of Independent Variables: If variables are independent, the variance of their sum is equal to the sum of their variances, allowing for straightforward calculations without needing covariances.

These properties simplify finding variance in composite functions involving multiple independent variables. They reduce complex calculations to manageable steps by utilizing the linearity and independence.

A linear combination involves combining multiple random variables, each multiplied by a constant, into a new random variable. In the given exercise, we see a linear combination expressed as \(Y = c_1 X_1 + c_2 X_2 + \cdots + c_p X_p\). Here, \(c_1, c_2, \ldots, c_p\) serve as weights applied to their respective variables \(X_1, X_2, \ldots, X_p\).When dealing with linear combinations, especially where your variables are independent, the main task is to calculate the variance of the new composite variable. You're essentially trying to understand how the combined uncertainty from each \(X_i\) contributes to the total uncertainty of \(Y\). Using the variance properties mentioned earlier, because the \(X_i\)s are independent, you calculate the overall variance as follows:

• Find the variance of each scaled variable: \(V(c_i X_i) = c_i^2 V(X_i)\).
• Sum these variances: \(V(Y) = c_1^2 V(X_1) + c_2^2 V(X_2) + \cdots + c_p^2 V(X_p)\).

These calculations highlight how the variance from each weighted variable contributes additively to the aggregate variance, simplifying problems into concise solutions.