Normalization of a probability distribution ($\int f(x)dx=1$)

A probability distribution assigns total probability one to its sample space. For a nonnegative probability density function this means that its integral over the full domain equals one; for a probability mass function the sum of its masses equals one. A nonnegative integrable function with positive finite integral can be divided by that integral to normalize it. A negative function cannot be a probability density function even if a formal manipulation purports to normalize it.