List of integrals of Gaussian functions

In these expressions

\phi(x) = \tfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}

is the standard normal probability density function, and

\textstyle \Phi(x) = \int_{-\infty}^x \phi(t)dt = \frac12\big(1 + \operatorname{erf}\big(\frac{x}{\sqrt{2}}\big)\big)

is the corresponding cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

(where erf is the error function

Error function

In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...

Indefinite integrals

list this integral without the minus sign, which is an error. See calculation by WolframAlpha

(in these integrals, n!! is the double factorial: for even n’s it is equal to the product of all even numbers from 2 to n, and for odd n’s it is the product of all odd numbers from 1 to n, additionally it is assumed that )

report this integral with error, see WolframAlpha

Definite integrals

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