Download - Central limit theorem
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πΏ π₯π
π (πΏ π₯ )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
π (π¦ππ )
0π¦ ππ
πΏ π¦β π (πΏ π¦ )π (πΏπ₯1 )|π΄ππΈ πΏπ₯1+β―
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Central limit theorem
π (π₯ )= 1π β2π
πβ 12 ( π₯βππ )
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HT
πΏ π₯π
π (πΏ π₯ )
0 1 2 3-1-2-3
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πΏ π₯π
π (πΏ π₯ )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
π (π¦ππ )
0π¦ ππ
πΏ π¦β π (πΏ π¦ )π (πΏπ₯1 )|π΄ππΈ πΏπ₯1+β―
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Physics lab: Engineered for tightly-controlled noise
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Physics lab: Engineered for tightly-controlled noise
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Physics lab: Engineered for tightly-controlled noise
t
V
x1
x2
x3
x5
x4
Hook vibration
Uneven air flow
Thermal expansion
Laser pointer vibration
Twisting
y
πΏ π¦=πΏ π¦ (πΏπ₯1 , πΏπ₯2 , πΏπ₯3 ,β― )
πΏ π¦β πΏ π¦ π΄ππΈ+π (πΏ π¦ )π (πΏ π₯1 )|π΄ππΈπΏπ₯1+ π (πΏ π¦ )
π (πΏπ₯2 )|π΄ππΈπΏπ₯2+β―π
π 1 π 2
βSmallβ noise: Neglect quadratic terms in Taylor expansion
0
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πΏ π₯π
π (πΏ π₯ )
0 1-1
Central limit theorem
Gaussians everywhere
Gaussians in physics
Slightly-disguised Gaussians in biology
π (π¦ππ )
0π¦ ππ
πΏ π¦β π (πΏ π¦ )π (πΏπ₯1 )|π΄ππΈ πΏπ₯1+β―
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π π¦ππ‘
=π π¦
ππ +ΒΏππ +ΒΏ
ππ‘+ π π¦ππ β
π π β
ππ‘ΒΏΒΏ
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Biology: Law of mass action and logarithms
yx2x1 x3
y
y yy
y yy y y
π+ΒΏ ΒΏ πβπ +ΒΏΒΏ π β
π+ΒΏ π₯1π₯2π₯3β― ΒΏ πβπ¦+1 -1
0=π+ΒΏ π₯1π₯2 π₯3β―βπβ π¦ππ ΒΏ
πβπ¦ ππ=π+ΒΏπ₯1π₯2 π₯3β― ΒΏ
ln ( π¦ππ )=ln ΒΏΒΏFluctuations in x1, x2, x3, etc. are not necessarily engineered to be small. First-order Taylor-expansion might be inaccurate.
ln ( π¦ππ )=ln ΒΏΒΏln ( π¦ππ )β ln ΒΏΒΏ
π π 2π 1
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Biology: Law of mass action and logarithms
ln ( π¦ππ )β ln ΒΏΒΏπ π 2π 1
A histogram of the logarithm of the concentration of y displays a normal distribution
30001507 3001500
π [ln ( π¦ππ ) ]
5 6 7 82 3 4ln ( π¦ππ )
π (π¦ππ )
200 3001000 π¦ ππ
yST = e4 = 55
e5 = 150
e6 = 400