Final answer:
Continuous probability functions assign a zero probability to any exact value. Integrals of these functions over an interval give the probability of a random variable falling within that interval. For a constant function within a range, the integral is the product of the constant and the range.
Step-by-step explanation:
This question pertains to the mathematical concept of integration within the context of continuous probability functions. When we are dealing with a continuous probability distribution, the probability of a specific value happening is zero because there are an infinite number of possible outcomes in a continuum.
Therefore, for questions like what is P(x > 15) or P(x = 7), the answer is straightforward: it is 0, as the probability of any exact value in a continuous distribution is 0.
We also address the concept that, for continuous functions, the integral represents the area under the curve. Therefore, the integral of f(x) from one point to another, in the context of a probability density function (pdf), is equal to the probability of a random variable falling within that interval.
For instance, if f(x) is represented by a constant, such as 12 within the interval 0 to 12, then to find P(0 < x < 12) we'd integrate f(x) over the interval, which in this case simply equates to the product of the constant and the interval length because the graph is a horizontal line. As long as the pdf is normalized (total area is 1), the integral over the entire support of the variable will sum to 1.