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Convergence_in_distribution_(sum_of_uniform_rvs).gif (200 × 148 pixels, file size: 20 KB, MIME type: image/gif, looped, 9 frames, 12 s)
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Summary
| Description |
Z_n is a normalized sum of iid uniform random variables: Z_n = 1/√n Sum{U(-1,1): i=1,...,n}. The animation shows how the pdfs of Z_n converge to a normal N(0,⅓) random variable. |
| Source | Own work |
| Author | Stpasha |
Mathematica source
f[x_] := If[-1 <= x <= 1, 1/2, 0];
f2[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f[x - t] \[DifferentialD]t\)];
f3[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f2[x - t] \[DifferentialD]t\)];
f4[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f3[x - t] \[DifferentialD]t\)];
f5[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f4[x - t] \[DifferentialD]t\)];
f6[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f5[x - t] \[DifferentialD]t\)];
f7[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f6[x - t] \[DifferentialD]t\)];
f8[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f7[x - t] \[DifferentialD]t\)];
f9[x_] := Evaluate[\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]f[t] f8[x - t] \[DifferentialD]t\)];
fn[n_, x_] := \[Piecewise] {
{Sqrt[1] f1[Sqrt[1] x], n == 1},
{Sqrt[2] f2[Sqrt[2] x], n == 2},
{Sqrt[3] f3[Sqrt[3] x], n == 3},
{Sqrt[4] f4[Sqrt[4] x], n == 4},
{Sqrt[5] f5[Sqrt[5] x], n == 5},
{Sqrt[6] f6[Sqrt[6] x], n == 6},
{Sqrt[7] f7[Sqrt[7] x], n == 7},
{Sqrt[8] f8[Sqrt[8] x], n == 8},
{Sqrt[9] f9[Sqrt[9] x], n == 9}
}
Table[
Plot[fn[n, x], {x, -2, 2},
Exclusions -> None,
PlotRange -> {0, 0.8},
ImageSize -> 200,
PlotStyle -> Thickness[Large],
LabelStyle -> Directive[Larger],
Epilog -> Inset[Style["\!\(\*StyleBox[\"n\",\nFontSlant->\"Italic\"]\) = " <> ToString[n],18], {1.2, 0.75}]
],
{n, 1, 9, 1}
]
Export["c:/anim.gif", %,
"DisplayDurations" -> {1, 1, 1, 1, 1, 1, 1, 1, .25},
"TransparentColor" -> White
]
Licensing
 
|
This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
|
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 23:50, 13 September 2009 | | 200 × 148 (20 KB) | Stpasha | {{Information |Description={{en|1=Z_n is a normalized sum of iid uniform random variables: Z_n = 1/√n Sum[][-1,1], i=1,...,n]. The animation shows how the pdfs of Z_n converge to a normal N(0, ⅓) random variable.}} |Source=Own work by uploader |A |
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