Final answer:
The unconstrained minimum of f(x) = E(Y − x)² is found when x equals the mean (μ) of the discrete real random variable Y. The expected value is calculated by summing the products of each realization of Y with its probability, and the minimum of this expectation function corresponds to the mean of the distribution.
Step-by-step explanation:
The goal is to find the unconstrained minimum of the function f(x) = E(Y − x)², which represents the expected value of the squared difference between a discrete real random variable Y and a constant x. To calculate the expected value, one would multiply each realization of Y, denoted as yk, by its probability pk and sum these products across all n possible outcomes. The formula used here is E(X) = μ = Σ xP(x).
Minimizing f(x) essentially involves differentiating it with respect to x and finding the value of x for which the derivative equals zero. As a function of expectation, this minimum is conventionally found at the mean of the random variable Y. The function f(x) achieves its minimum when x equals the mean (μ) of Y, because the mean is the balance point of the distribution where the sum of the squared differences weighted by their probabilities is at its lowest.
The mean or expected value of Y is thus the critical value that minimizes f(x). This concept is fundamental in statistics when dealing with probability distribution functions and optimizing expected outcomes.