probability distributions: who cares & why? · another strange thing!-2 0 2 0 2 4 6 8 10...
TRANSCRIPT
![Page 1: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/1.jpg)
Probability distributions: Who cares & why?
Arnab Chakraborty
Indian Statistical Institute
Nov 12, 2017
![Page 2: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/2.jpg)
“Snakes and Ladders” ludo
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“Snakes and Ladders” ludo
![Page 4: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/4.jpg)
“Snakes and Ladders” ludo
![Page 5: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/5.jpg)
A strange ludo
1.Xnew = 0.8Xold + 0.1Ynew = 0.8Yold + 0.04
2.Xnew = 0.5Xold + 0.25Ynew = 0.5Yold + 0.4
3.Xnew = 0.355Xold − 0.355Yold +0.266Ynew = 0.355Xold +0.355Yold +0.078
4.Xnew = 0.355Xold + 0.355Yold + 0.378Ynew = −0.355Xold + 0.355Yold + 0.434
![Page 6: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/6.jpg)
A strange ludo
1.Xnew = 0.8Xold + 0.1Ynew = 0.8Yold + 0.04
2.Xnew = 0.5Xold + 0.25Ynew = 0.5Yold + 0.4
3.Xnew = 0.355Xold − 0.355Yold +0.266Ynew = 0.355Xold +0.355Yold +0.078
4.Xnew = 0.355Xold + 0.355Yold + 0.378Ynew = −0.355Xold + 0.355Yold + 0.434
![Page 7: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/7.jpg)
A strange ludo
1.Xnew = 0.8Xold + 0.1Ynew = 0.8Yold + 0.04
2.Xnew = 0.5Xold + 0.25Ynew = 0.5Yold + 0.4
3.Xnew = 0.355Xold − 0.355Yold +0.266Ynew = 0.355Xold +0.355Yold +0.078
4.Xnew = 0.355Xold + 0.355Yold + 0.378Ynew = −0.355Xold + 0.355Yold + 0.434
![Page 8: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/8.jpg)
A strange ludo
1.Xnew = 0.8Xold + 0.1Ynew = 0.8Yold + 0.04
2.Xnew = 0.5Xold + 0.25Ynew = 0.5Yold + 0.4
3.Xnew = 0.355Xold − 0.355Yold +0.266Ynew = 0.355Xold +0.355Yold +0.078
4.Xnew = 0.355Xold + 0.355Yold + 0.378Ynew = −0.355Xold + 0.355Yold + 0.434
![Page 9: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/9.jpg)
A strange thing!
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A strange thing!
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A strange thing!
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A strange thing!
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A strange thing!
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Try it out yourself!
https://arnab-chakraborty.shinyapps.io/shny/
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Another strange thing!
−2 0 2
02
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Statistical regularity
Regularity in randomness!
I Not always
I Only when we have “lots of randomness”
Natural phenomena:
I Leaves: very similar but not same
I Finger prints
![Page 17: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/17.jpg)
Statistical regularity
Regularity in randomness!
I Not always
I Only when we have “lots of randomness”
Natural phenomena:
I Leaves: very similar but not same
I Finger prints
![Page 18: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/18.jpg)
Why care?
1. Understanding: Probability
2. Using: Statistics
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Why care?
1. Understanding: Probability
2. Using: Statistics
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Why care?
1. Understanding: Probability
2. Using: Statistics
![Page 21: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/21.jpg)
Histogram
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Histogram
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Histogram
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Histogram
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Probability density function
0.0 0.2 0.4 0.6 0.8 1.0
01
23
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Probability density function
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0.0
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Probability density function
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0.0
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2.5
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Probability density function
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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Probability density function
Probabilitydensity function
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Probability density function
Probabilitydensity function
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0.0
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Probability density function
Probabilitydensity function
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0.0
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All continuous randomvariables have PDFs.
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Probability density function
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Probability density function
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Probability density function
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Probability density function
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Probability density function
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Probability density function
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Probability density function
All continuous randomvariable pairs have jointPDFs.
![Page 39: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/39.jpg)
Molecules
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Molecules
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A single molecule
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A single molecule
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A single molecule
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A single molecule
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A single molecule
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A single molecule
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Isotropy
x , y , z are continuous random variables and so havePDFs.
In case of ”no flow”
I They have the same density (call it f (·))I They are independent.
I So joint density of x , y , z is f (x)f (y)f (z).
I f (x)f (y)f (z) does not depends only on the length of(x , y , z) and not on the direction.
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Isotropy
x , y , z are continuous random variables and so havePDFs.
In case of ”no flow”
I They have the same density (call it f (·))
I They are independent.
I So joint density of x , y , z is f (x)f (y)f (z).
I f (x)f (y)f (z) does not depends only on the length of(x , y , z) and not on the direction.
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Isotropy
x , y , z are continuous random variables and so havePDFs.
In case of ”no flow”
I They have the same density (call it f (·))I They are independent.
I So joint density of x , y , z is f (x)f (y)f (z).
I f (x)f (y)f (z) does not depends only on the length of(x , y , z) and not on the direction.
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Isotropy
x , y , z are continuous random variables and so havePDFs.
In case of ”no flow”
I They have the same density (call it f (·))I They are independent.
I So joint density of x , y , z is f (x)f (y)f (z).
I f (x)f (y)f (z) does not depends only on the length of(x , y , z) and not on the direction.
![Page 51: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/51.jpg)
Isotropy
x , y , z are continuous random variables and so havePDFs.
In case of ”no flow”
I They have the same density (call it f (·))I They are independent.
I So joint density of x , y , z is f (x)f (y)f (z).
I f (x)f (y)f (z) does not depends only on the length of(x , y , z) and not on the direction.
![Page 52: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/52.jpg)
Isotropy
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Isotropy
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Isotropy
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Mathematically...
f (x)f (y)f (z) = g(x2 + y 2 + z2)
f ′(x)f (y)f (z) = 2xg ′(x2 + y 2 + z2)
f (x)f ′(y)f (z) = 2yg ′(x2 + y 2 + z2)
f (x)f (y)f ′(z) = 2zg ′(x2 + y 2 + z2)
g ′(x2+y 2+z2) =f ′(x)f (y)f (z)
2x=
f (x)f ′(y)f (z)
2y=
f (x)f (y)f ′(z)
2z.
f ′(x)
xf (x)=
f ′(y)
yf (y)=
f ′(z)
zf (z)
= k , say
.
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Mathematically...
f (x)f (y)f (z) = g(x2 + y 2 + z2)
f ′(x)f (y)f (z) = 2xg ′(x2 + y 2 + z2)
f (x)f ′(y)f (z) = 2yg ′(x2 + y 2 + z2)
f (x)f (y)f ′(z) = 2zg ′(x2 + y 2 + z2)
g ′(x2+y 2+z2) =f ′(x)f (y)f (z)
2x=
f (x)f ′(y)f (z)
2y=
f (x)f (y)f ′(z)
2z.
f ′(x)
xf (x)=
f ′(y)
yf (y)=
f ′(z)
zf (z)
= k , say
.
![Page 57: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/57.jpg)
Mathematically...
f (x)f (y)f (z) = g(x2 + y 2 + z2)
f ′(x)f (y)f (z) = 2xg ′(x2 + y 2 + z2)
f (x)f ′(y)f (z) = 2yg ′(x2 + y 2 + z2)
f (x)f (y)f ′(z) = 2zg ′(x2 + y 2 + z2)
g ′(x2+y 2+z2) =f ′(x)f (y)f (z)
2x=
f (x)f ′(y)f (z)
2y=
f (x)f (y)f ′(z)
2z.
f ′(x)
xf (x)=
f ′(y)
yf (y)=
f ′(z)
zf (z)
= k , say
.
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Mathematically...
f (x)f (y)f (z) = g(x2 + y 2 + z2)
f ′(x)f (y)f (z) = 2xg ′(x2 + y 2 + z2)
f (x)f ′(y)f (z) = 2yg ′(x2 + y 2 + z2)
f (x)f (y)f ′(z) = 2zg ′(x2 + y 2 + z2)
g ′(x2+y 2+z2) =f ′(x)f (y)f (z)
2x=
f (x)f ′(y)f (z)
2y=
f (x)f (y)f ′(z)
2z.
f ′(x)
xf (x)=
f ′(y)
yf (y)=
f ′(z)
zf (z)
= k , say
.
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Mathematically...
f (x)f (y)f (z) = g(x2 + y 2 + z2)
f ′(x)f (y)f (z) = 2xg ′(x2 + y 2 + z2)
f (x)f ′(y)f (z) = 2yg ′(x2 + y 2 + z2)
f (x)f (y)f ′(z) = 2zg ′(x2 + y 2 + z2)
g ′(x2+y 2+z2) =f ′(x)f (y)f (z)
2x=
f (x)f ′(y)f (z)
2y=
f (x)f (y)f ′(z)
2z.
f ′(x)
xf (x)=
f ′(y)
yf (y)=
f ′(z)
zf (z)= k , say.
![Page 60: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/60.jpg)
Solving
df
dx= kxf .
df
f= kxdx .∫
df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.
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Solving
df
dx= kxf .
df
f= kxdx .
∫df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.
![Page 62: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/62.jpg)
Solving
df
dx= kxf .
df
f= kxdx .∫
df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.
![Page 63: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/63.jpg)
Solving
df
dx= kxf .
df
f= kxdx .∫
df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.
![Page 64: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/64.jpg)
Solving
df
dx= kxf .
df
f= kxdx .∫
df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.
![Page 65: Probability distributions: Who cares & why? · Another strange thing!-2 0 2 0 2 4 6 8 10 100000. Statistical regularity Regularity in randomness! I Not always I Only when we have](https://reader033.vdocuments.net/reader033/viewer/2022050412/5f8938627f414876ec24a41c/html5/thumbnails/65.jpg)
Solving
df
dx= kxf .
df
f= kxdx .∫
df
f= k
∫xdx .
log f =kx2
2+ const.
f = const × ekx2
2 .
Maxwell / Gaussian distribution.