What is the area under f(x) if the function is a continuous probability density function?

The area under a continuous probability density function (pdf), f(x), over its entire space is equal to 1. This is a fundamental property of pdfs in probability theory.

Here's why:

1. The total probability of all outcomes (events) of a random variable must add up to 1. This is a basic axiom of probability.

2. For a continuous random variable, the probability of any specific outcome is defined as the area under the pdf over that outcome.

3. Therefore, the total area under the pdf over all possible outcomes (i.e., the entire space of the random variable) must add up to the total probability, which is 1.

So, if f(x) is a continuous pdf, the area under f(x) over its entire space is 1.