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Is there a name and formula for this curve? It isn't the “normal distribution”

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0

$begingroup$

1. starts at -1,0 asymptotically to x axis


2. maxes out at 0,1


3. and back down again to mirror how it started.

Normal distributions keep tapering off well past the -1..1 endpoints, and I was looking for something simple that ended. And most importantly the formula for it!

curves

share|cite|improve this question

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  There's many different such functions. For instance, $(x^2-1)^2$ and $(cos(pi x)+1)/2$
  $endgroup$
  – Calvin Khor
  Mar 29 at 3:15
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Search for 'bump functions' on wikipedia.
  $endgroup$
  – D.B.
  Mar 29 at 3:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Looks like the "truncated normal distribution".
  $endgroup$
  – mjw
  Mar 29 at 3:22
  

  
  
  
  

add a comment | 

0

$begingroup$

1. starts at -1,0 asymptotically to x axis


2. maxes out at 0,1


3. and back down again to mirror how it started.

Normal distributions keep tapering off well past the -1..1 endpoints, and I was looking for something simple that ended. And most importantly the formula for it!

curves

share|cite|improve this question

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  There's many different such functions. For instance, $(x^2-1)^2$ and $(cos(pi x)+1)/2$
  $endgroup$
  – Calvin Khor
  Mar 29 at 3:15
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Search for 'bump functions' on wikipedia.
  $endgroup$
  – D.B.
  Mar 29 at 3:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Looks like the "truncated normal distribution".
  $endgroup$
  – mjw
  Mar 29 at 3:22
  

  
  
  
  

add a comment | 

$begingroup$

1. starts at -1,0 asymptotically to x axis


2. maxes out at 0,1


3. and back down again to mirror how it started.

Normal distributions keep tapering off well past the -1..1 endpoints, and I was looking for something simple that ended. And most importantly the formula for it!

curves

share|cite|improve this question

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

$endgroup$

share|cite|improve this question

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

share|cite|improve this question

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

asked Mar 29 at 3:12

Benjamin HBenjamin H

1135

2 Answers
2

active

oldest

votes

2

$begingroup$

Your conditions do not single out a particular function. For instance, $(x^2-1)^2, fraccos(pi x)+1)2$ work. The first is a polynomial so that could be nice. The second is trigonometric.

If you want one that extends by 0 to a $C^infty(mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $exp(-1/(1-x^2))$.

Lets say $mathcal F$ is the collection of functions $:[0,1]to mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:

• 
  $mathcal F$ is convex: $f,ginmathcal F$ and $lambdain[0,1]$ implies $lambda f + (1-lambda) g in mathcal F$.


• if $fin mathcal F$, and $s ge 1$, then $f^sinmathcal F$.

Here's some graphs on Desmos:

share|cite|improve this answer

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

$endgroup$

add a comment | 

0

$begingroup$

The curves you desire are a subset the Superparabola. These are specifically in domain $xin[-1,1]$. It is also parameterized so that you can have smooth transition between the functions.

share|cite|improve this answer

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

$endgroup$

add a comment | 

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2

$begingroup$

Your conditions do not single out a particular function. For instance, $(x^2-1)^2, fraccos(pi x)+1)2$ work. The first is a polynomial so that could be nice. The second is trigonometric.

If you want one that extends by 0 to a $C^infty(mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $exp(-1/(1-x^2))$.

Lets say $mathcal F$ is the collection of functions $:[0,1]to mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:

• 
  $mathcal F$ is convex: $f,ginmathcal F$ and $lambdain[0,1]$ implies $lambda f + (1-lambda) g in mathcal F$.


• if $fin mathcal F$, and $s ge 1$, then $f^sinmathcal F$.

Here's some graphs on Desmos:

share|cite|improve this answer

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

$endgroup$

add a comment | 

2

$begingroup$

Your conditions do not single out a particular function. For instance, $(x^2-1)^2, fraccos(pi x)+1)2$ work. The first is a polynomial so that could be nice. The second is trigonometric.

If you want one that extends by 0 to a $C^infty(mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $exp(-1/(1-x^2))$.

Lets say $mathcal F$ is the collection of functions $:[0,1]to mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:

• 
  $mathcal F$ is convex: $f,ginmathcal F$ and $lambdain[0,1]$ implies $lambda f + (1-lambda) g in mathcal F$.


• if $fin mathcal F$, and $s ge 1$, then $f^sinmathcal F$.

Here's some graphs on Desmos:

share|cite|improve this answer

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

$endgroup$

add a comment | 

$begingroup$

Your conditions do not single out a particular function. For instance, $(x^2-1)^2, fraccos(pi x)+1)2$ work. The first is a polynomial so that could be nice. The second is trigonometric.

If you want one that extends by 0 to a $C^infty(mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $exp(-1/(1-x^2))$.

Lets say $mathcal F$ is the collection of functions $:[0,1]to mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:

• 
  $mathcal F$ is convex: $f,ginmathcal F$ and $lambdain[0,1]$ implies $lambda f + (1-lambda) g in mathcal F$.


• if $fin mathcal F$, and $s ge 1$, then $f^sinmathcal F$.

Here's some graphs on Desmos:

share|cite|improve this answer

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

$endgroup$

share|cite|improve this answer

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

answered Mar 29 at 3:35

Calvin KhorCalvin Khor

12.5k21439

0

$begingroup$

The curves you desire are a subset the Superparabola. These are specifically in domain $xin[-1,1]$. It is also parameterized so that you can have smooth transition between the functions.

share|cite|improve this answer

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

$endgroup$

add a comment | 

0

$begingroup$

The curves you desire are a subset the Superparabola. These are specifically in domain $xin[-1,1]$. It is also parameterized so that you can have smooth transition between the functions.

share|cite|improve this answer

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

$endgroup$

add a comment | 

$begingroup$

The curves you desire are a subset the Superparabola. These are specifically in domain $xin[-1,1]$. It is also parameterized so that you can have smooth transition between the functions.

share|cite|improve this answer

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

$endgroup$

share|cite|improve this answer

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623

answered Mar 29 at 20:36

Cye WaldmanCye Waldman

4,2222623