According to the properties of variance, when a random variable Y is multiplied by a constant k, the variance of the new variable kY is equal to k squared multiplied by the variance of the original variable Y. This demonstrates that scaling a variable by k increases its variance by a factor of k squared.

According to the properties of variance, Var(aX + b) = a^2 * Var(X). Here, a = 2 and b = 3. Therefore, Var(2X + 3) = 2^2 * Var(X) = 4 * Var(X). The constant term b does not affect the variance because shifting the data does not change the spread.

Expected profit is calculated by taking the sum of all possible profit outcomes, each multiplied by its respective probability of occurrence. This represents the long-term average profit one would expect if the scenario were repeated many times.

The mean of a hypergeometric distribution, where N is the population size, K is the number of successes in the population, and n is the sample size, is given by n multiplied by (K/N). This represents the expected number of successes in the sample drawn without replacement.

The expected value, or mean, of a discrete random variable X is defined as the weighted average of all possible values that X can take, where the weights are the probabilities of those values. Mathematically, this is expressed as the sum of the product of each value x and its corresponding probability P(x), denoted as E(X) = ∑ x * P(x).

The expected value of the deviation of a random variable from its mean is defined as E[X - E(X)]. By the linearity of expectation, this becomes E(X) - E(E(X)). Since E(X) is a constant, E(E(X)) = E(X), resulting in E(X) - E(X) = 0. This property reflects that the mean is the center of gravity of the distribution.

The variance of a constant 'a' is zero. For a linear transformation (a + bX), the variance is calculated as Var(a + bX) = Var(a) + Var(bX). Since Var(a) = 0 and Var(bX) = b^2 * Var(X), the resulting variance is b^2 * Var(X). This property demonstrates that variance is invariant to shifts in origin but sensitive to changes in scale.

The variance is defined as E[(X - E[X])^2]. The provided answer D appears to be a typographical representation of E[(X - μ)^2], where μ is the mean. We preserve the answer key while noting the likely notation error in the source material.

To calculate the expected profit for each outcome, multiply the demand by the profit per unit ($0.50) and then by the probability of that demand occurring. For 21 units: 21 * 0.50 * 0.29 = 3.045. For 19 units: 19 * 0.50 * 0.40 = 3.8. For 22 units: 22 * 0.50 * 0.35 = 3.85.

The expected value is the long-run average or mean value of a random variable over many repeated trials. It is calculated by summing the products of each possible outcome and its associated probability, representing the theoretical center of a probability distribution.