Looking for features $f$ with $\int_{-\infty}^{\infty}f(x)\,dx = 1$.

I am seeking features and/or constants that when being incorporated from minus infinity to infinity generate 1. I assume the Dirac delta function is one instance yet probably there are some even more? Referrals on valuable product is additionally substantially valued.

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2019-05-07 13:35:34
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Answers: 7

One example is the standard Gaussian distribution, $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$. This is one of the most uncomplicated instance of a continual probability circulation function as stated by KennyTM over.

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2019-05-09 09:04:32
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The function which is 1 on the interval [0;1], and also 0 in other places, is a non - continual probability circulation function. The function which is 3 on [0;1] and also - 1 on (1 ;3 ], and more and also on. What sort of solution do you desire? What sort of buildings do you desire your features to have?

There actually are way too many features to checklist, given that increasing any kind of function by a C ^ oo function with portable assistance and afterwards using Kenny's method offers you a solution.

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2019-05-09 09:03:26
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This need to be a comment yet I can not comment ...

The Dirac delta function is not a function!

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2019-05-09 09:02:33
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Most sensible mother wavelet have square standard 1.

$\int_{-\infty}^\infty |\psi(t)|^2 dt = 1$

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2019-05-09 09:02:17
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If you take any kind of weird function $F$ differentiable on $\mathbb{R}$ and also such that $F(x)\to l$ (with $l$ a nonzero actual) for $x\to \infty$, after that $f(x) = \frac{1}{2l}F'(x)$ statisfies your demand. As an example $F(x) = \operatorname{arctan}(x)$

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2019-05-09 08:41:56
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Any integrable features that offers a limited nonzero solution can be changed to match your demand.

Intend $\int f(x)dx=A$, after that allow $g(x)=f(x)/A$, instantly we have $\int g(x) dx=A/A=1$.

(Actually, all continual probability distribution function has to have this building.)

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2019-05-09 08:23:06
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Any function $f(x)$ which incorporates to $1$ over any kind of array $[a,b]$ fits this costs, given that we can specify $g(x)=f(x)$ on $[a,b]$, and also $0$ almost everywhere else.

Also if you just desire continual features, limiting ourselves over to $f(x)$ where $f(a)=f(b)=0$ still pleases this.

If you desire continual features purely $>0$ almost everywhere, these are called chance circulations (continual on $[-\infty,\infty]$). A huge checklist of such features can be located here A couple of even more remarkable instances are:

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2019-05-09 08:22:41
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