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I have this probability distribution:

$$f(x) = a\exp\left(-\frac{(x-b)^2}{2c^2}\right)$$

I know constant $b = 0, c = 9.49$ and area under the curve is equal to $1$. How to calculate constant $a$ when I can't integrate the function?

StubbornAtom
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andz
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  • why can't you integrate? – user619894 Oct 13 '20 at 18:33
  • There is no closed form antiderivative for a Gaussian, but one can evaluate the indefinite integral $\int_{\mathbb{R}} f$. This is a pretty standard exercise, and has been answered on Math SE before. See, for example, https://math.stackexchange.com/questions/1125233/show-that-integral-of-gaussian-distribution-is-1 . – Xander Henderson Oct 18 '20 at 15:00

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Using the substitution $u=\frac{x-b}{\sqrt2 c}$, you obtain: $$1=a\int_{-\infty}^\infty\exp\left(-\frac{(x-b)^2}{2c^2}\right)\,dx=\sqrt{2}ac\int_{-\infty}^{\infty}\exp\left(-u^2\right)\,du=\sqrt{2\pi}ac.$$ Now you can solve for $a$ since you know the value of $c$.

csch2
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