Normalization constant

The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.

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 a function ${\displaystyle \varphi (x)}$ as

so that

then the function ${\displaystyle \varphi (x)}$ 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 ${\displaystyle {\frac {1}{\sqrt {2\pi \,}}}}$ is the normalizing constant of function ${\displaystyle p(x)}$.

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