Applications of Integration to Probability Density Functions

Continuous probability distributions model random variables that can take any real value within a range. For example, the height of adult females might be 163.5, 165.25 centimeters, or any value in between. This makes height a continuous random variable.

The probability of this variable is described using a probability density function—a smooth curve over the variable’s range. The unit of this density is the reciprocal of the variable’s unit.

The total probability that the variable falls within a specific interval is found by calculating the area under the curve for that range. For example, the probability that a woman’s height is between 150 and 170 centimeters is found by integrating the density function over that range.

If the result is 0.75, it means that 75% of women in the population have heights within that interval.

But for an exact height, the probability is zero because integrating over a single point gives zero area.

A valid probability density function is always non-negative and integrates to 1 across the entire range of possible values for the random variable.