Area under a guassian with two known points using error function

Question

I am interested in a nice function to find the area of a section of a normalized Gaussian distribution. Let's say I have a normalized Gaussian distribution of standard deviation $\sigma$, mean $\mu$, and I am interested in the area centered at $\mu$ between +/- $t$ ( see below)

Is there a simple way to express this for any given $t$ using the erf? The ref from which I found the above figure uses

$$a_1 = 2 \mathrm{erf}\left(\frac{t}{\sigma} \right) $$

But I don't think this expression is true for a normalized guassian. Moreover, can the $\textit{total}$ area be expressed by an error function?

Thanks, very new to statistics.

Which quantities in the diagram are 'known'? – Glen_b Nov 17 '19 at 02:41 — Glen_b, Nov 17 '19 at 02:41
all of them- t, sigma, and h are all known. – Jimbo Nov 18 '19 at 05:38 — Jimbo, Nov 18 '19 at 05:38

score 0 · Answer 1 · answered Nov 08 '19 at 10:07

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This question has already been answered here. Note that you can compute the integral over a finite interval by means of the CDF $\Phi$: $$\int_a^b \varphi(x)\,dx = \Phi(b) - \Phi(a)$$

answered Nov 08 '19 at 10:07

cdalitz

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Area under a guassian with two known points using error function

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