Matching widths of two different functions
0
$begingroup$
Say I have a rectangular function as follows $$f(x) =
begin{cases}
1 & text{$|x|$$leq$$x_0$} \[2ex]
0 & text{otherwise}
end{cases}$$
I want to match its width (which in this case is $2x_0$) to that of a Gaussian defined as, say, $exp(-2x^2/sigma^2)$ (i.e., $sigma^2$. I want to write some relation between $sigma^2$ and $x_0$). Is there a standard or conventional way to do this?
integration gaussian-integral rectangles
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
$endgroup$
add a comment |
0
$begingroup$
Say I have a rectangular function as follows $$f(x) =
begin{cases}
1 & text{$|x|$$leq$$x_0$} \[2ex]
0 & text{otherwise}
end{cases}$$
I want to match its width (which in this case is $2x_0$) to that of a Gaussian defined as, say, $exp(-2x^2/sigma^2)$ (i.e., $sigma^2$. I want to write some relation between $sigma^2$ and $x_0$). Is there a standard or conventional way to do this?
integration gaussian-integral rectangles
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
$endgroup$
$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47
add a comment |
0
0
0
$begingroup$
Say I have a rectangular function as follows $$f(x) =
begin{cases}
1 & text{$|x|$$leq$$x_0$} \[2ex]
0 & text{otherwise}
end{cases}$$
I want to match its width (which in this case is $2x_0$) to that of a Gaussian defined as, say, $exp(-2x^2/sigma^2)$ (i.e., $sigma^2$. I want to write some relation between $sigma^2$ and $x_0$). Is there a standard or conventional way to do this?
integration gaussian-integral rectangles
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
$endgroup$
Say I have a rectangular function as follows $$f(x) =
begin{cases}
1 & text{$|x|$$leq$$x_0$} \[2ex]
0 & text{otherwise}
end{cases}$$
I want to match its width (which in this case is $2x_0$) to that of a Gaussian defined as, say, $exp(-2x^2/sigma^2)$ (i.e., $sigma^2$. I want to write some relation between $sigma^2$ and $x_0$). Is there a standard or conventional way to do this?
integration gaussian-integral rectangles
integration gaussian-integral rectangles
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
asked Dec 20 '18 at 10:30
FlyGuyFlyGuy
13
13
$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47
add a comment |
$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47
$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47
add a comment |
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$begingroup$
A Gaussian is nonzero everywhere, so it has no width in the sense $f(x)$ has. On the other hand, the width of the bell curve is defined by the value of $sigma$. For example, $99.5$% of its surface area is under the curve on the interval $[-2sigma , 2sigma]$.
$endgroup$
– User123456789
Dec 20 '18 at 10:39
$begingroup$
There's no doubt that you're right. I'll try to explain myself a little better: In some calculations I made I assumed $f(x)$ to be a Gaussian function. Now, in some scenarios this assumption doesn't hold and $f(x)$ could take the form of a rectangular function. I was looking for a way to match the $sigma^2$ from the Gaussian I assumed to that of a width of a rectangular function. I realize that widths here mean something different for each function and the result wouldn't really be accurate.
$endgroup$
– FlyGuy
Dec 20 '18 at 10:43
$begingroup$
anyone with any ideas?
$endgroup$
– FlyGuy
Dec 22 '18 at 14:47