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

Question

I am looking for functions and/or constants that when being integrated from minus infinity to infinity produce 1. I think the Dirac delta function is one example but perhaps there are some more? References on useful material is also greatly appreciated.

Answer 1

Any integrable functions that gives a finite nonzero answer can be modified to suit your need.

Suppose ∫ f(x) dx = A, then let g(x) = f(x)/A, automatically we have ∫ g(x) dx = A/A = 1.

(Actually, all continuous probability distribution function must have this property.)

Answer 2

If you take any odd function $F$ differentiable on $\mathbb{R}$ and such that $F(x)\to l$ (with $l$ a nonzero real) for $x\to \infty$, then $f(x) = \frac{1}{2l}F'(x)$ statisfies your request. For example $F(x) = \operatorname{arctan}(x)$

Answer 3

Any function f(x) which integrates to 1 over any range [a,b] fits this bill, since we can define g(x) = f(x) on [a,b], and 0 everywhere else.

Even if you only want continuous functions, restricting ourselves above to f(x) where f(a) = f(b) = 0 still satisfies this.

If you want continuous functions strictly > 0 everywhere, these are known as probability distributions (continuous on [-∞, ∞]). A large list of such functions can be found here. A few more notable examples are:

• The normal distribution
• The skew-normal distribution
• The t-distribution
• The cauchy distribution
• The extreme-value distribution

Answer 4

Most practical mother wavelet have square norm 1.

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

Answer 5

This should be a comment but I cannot comment...

The Dirac delta function is not a function!

Answer 6

The function which is 1 on the interval [0;1], and 0 elsewhere, is a non-continuous probability distribution function. The function which is 3 on [0;1] and -1 on (1;3], and so on and on. What kind of answer do you want? What kind of properties do you want your functions to have?

There really are too many functions to list, since multiplying any function by a C^oo function with compact support and then applying Kenny's trick gives you an answer.

Answer 7

One good example is the standard Gaussian distribution, $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$. This is the most straightforward example of a continuous probability distribution function as mentioned by KennyTM above.