# Cumulative Probability Function and the Central Limit Theorem

Learn about cumulative probability, the Gaussian error function, central limit theorem, and how to measure the difference between distributions.

## Gaussian error function and cumulative probability (density) function

We have mainly discussed probabilities of single values as specified by the probability (density) functions. However, in many cases, we need to know the probabilities of having values within a certain range. The probability of a specific value of a continuous random variable is actually infinitesimally small (nearly zero), and only the probability of a range of values is finite and has a useful meaning of a probability. This integrated version of a probability density function is the probability of having a value $x$ for the random variable $X$ in the range of $x_1 \leq x \leq x_2$ and is given by the following equation:

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