Normalizing Constant - Definition and Examples

Definition and Examples

In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function. For example, if we define

we have

if we define function as

so that

Function is a probability density function. This is the density of the standard normal distribution. (Standard, in this case, means the expected value is 0 and the variance is 1.)

And constant is the normalizing constant of function .

Similarly,

and consequently

is a probability mass function on the set of all nonnegative integers. This is the probability mass function of the Poisson distribution with expected value λ.

Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. In that context, the normalizing constant is called the partition function.