Probability of Random Error under Gaussian Distribution occurring within Interval

Theorem

Let $X$ be a random variable with a Gaussian distribution.

Let:

the expectation of $X$ be $0$

the standard deviation of $X$ be $\sigma$.

Let $x \in \R$ be a real number.

Then the probability that a random error from $X$ lies in the interval $\openint {-x} x$ is given by:

$p = \map \erf {\dfrac x {\sigma \sqrt 2} }$

where $\erf$ denotes the error function.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error function