Probability of Random Error under Gaussian Distribution occurring within Interval
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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
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Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error function