Final answer:
To answer the question, we can show that the probability that g(x) is greater than or equal to k is less than the expected value of g(x) by utilizing the definitions of probability and expected value and their respective properties.
Step-by-step explanation:
The question concerns the demonstration of an inequality involving a probability statement and an expected value. Specifically, we must show that P[g(x) ≥ k] < E[g(x)], where P denotes probability, E denotes expected value, g(x) is a nonnegative function, and k is a constant. The inequality suggests that the probability of the function g(x) being greater than or equal to a constant k is less than the expected value of that function.
By definition, the expected value of a nonnegative random variable, like g(x), is E[g(x)] = ∫ g(x)f(x)dx, where f(x) is the probability density function of x. Inequality can be shown using the properties of expected value and probability. Specifically, we can use the property that the sum of probabilities over all possible outcomes of a random variable equals one, and the fact that the expected value is a weighted average of all possible values the function g(x) can take, weighted by their respective probabilities.