Definite integrals are really important in probability theory. They help us understand the area under curves when looking at data. Let’s break down how this works:

1. Probability Density Functions (PDFs):

   • A probability density function, or PDF, is like a map that shows how likely it is for a continuous random variable to occur at a certain number.
   • The total area under a PDF curve adds up to 1. This means there's a 100% chance that the random variable will fall within the range we are looking at.
   • You can think of it like this: $\int_{-\infty}^{\infty} f(x) \, dx = 1$

2. Finding Probabilities:

   • If we want to find the chance that a random variable $X$ lands between two values, $a$ and $b$, we can use definite integrals: $P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$

3. Example Use Case:

   • Take the normal distribution, which is common in statistics. If we know the average (mean) and how spread out the values are (standard deviation), we can find the probabilities of different situations—like test scores or measuring errors—by looking at the area under the curve.

In short, definite integrals help us calculate probabilities in continuous data. They are essential for understanding and analyzing statistics in various situations.