Applications of Integration to Probability Density Functions

Continuous probability distributions are used to model random variables that can take on any real value within a specified range. These variables do not take on isolated or countable values but rather exist on a continuum. For example, the height of an individual can be measured with increasing precision—such as 163.5 or 165.25 centimeters—demonstrating that height is a continuous random variable.

The behavior of such variables is described using a probability density function (PDF), which defines how probabilities are distributed over the range of possible values. Unlike discrete distributions, where individual outcomes have specific probabilities, continuous distributions assign probabilities to intervals of values. The total probability that a variable falls within a specific interval is represented by the area under the PDF curve over that range. This is calculated using integration:

For example, if the area under the curve between 150 and 170 centimeters is 0.75, then there is a 75% chance that a randomly selected individual has a height within that interval. However, the probability of the variable taking on any single exact value is always zero, as the area under a single point is zero.

To be valid, a probability density function must be non-negative for all values and must integrate to 1 over the entire domain of the variable. These properties ensure that the function accurately represents a probability distribution and that the total probability across all possible outcomes is equal to 1.