By the linearity property of expectation, the expected value of a linear combination of random variables is the linear combination of their expected values. Therefore, E(X1 - X2) = E(X1) - E(X2). If we denote the population means as μ1 and μ2, the result is μ1 - μ2. This is a fundamental property used in comparing means between two groups.

When two random variables are independent, the variance of their sum or their difference is equal to the sum of their individual variances. This property holds because the covariance between independent variables is zero, simplifying the general formula for the variance of a sum.

The expression E(X-µ)^2 represents the variance of a random variable X. In the context of a normal distribution, this value is denoted by σ². It quantifies the average squared deviation of the data points from the population mean, serving as a fundamental measure of the distribution's variability.

The expected value E(X) is defined as the weighted average of all possible values of a random variable, which is equivalent to the arithmetic mean. The provided answer key suggests 'Geometric Mean' (Option B), which is generally incorrect for the definition of expected value. We note this as a potential conflict.

The expected value of a constant is simply the constant itself. However, the provided answer key indicates '1' (Option C). This contradicts the standard statistical definition where E(C) = C. We must note this as a potential error in the source material.

Wald's Equation is a statistical concept that describes the expected value of a function of a random process. It states that the expected value of a function of a random process is equal to the expected value of the process times the derivative of the function with respect to the process.

The linearity of expectation states that E(ax + b) = aE(x) + b. Given the provided answer key is B, it appears to assume 'a' is 1. However, mathematically, the constant 'b' should be added. We retain the provided answer key despite the potential simplification or notation ambiguity.

The linearity of expectation states E(X+Y) = E(X) + E(Y). The provided answer B is mathematically incorrect as it suggests E(2X). We retain the answer key provided by the source while noting the conflict with standard statistical theory.

The linearity property of the expectation operator allows constants to be pulled out of the expectation. Therefore, for any constant 'a' and random variable 'x', the expected value E(ax) is equal to 'a' multiplied by the expected value of 'x', denoted as aE(x).

The variance of a linear transformation of a random variable is given by Var(a + bX). Since adding a constant 'a' does not change the spread of the distribution, Var(a + bX) = Var(bX). By the properties of variance, multiplying by a constant 'b' results in the variance being multiplied by the square of that constant, yielding b^2 * Var(X).