introduction to probabilitybiophysics/pc2267/lecture-02-prob.pdfoutline discrete distribution...
TRANSCRIPT
![Page 1: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/1.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Introduction to Probability
Jie Yan
August 18, 2006
Jie Yan Introduction to Probability
![Page 2: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/2.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Contents
I Discrete distribution.
I Continuous distribution.
I Important distributions
I Application: 1-d diffusion
Jie Yan Introduction to Probability
![Page 3: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/3.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Contents
I Discrete distribution.
I Continuous distribution.
I Important distributions
I Application: 1-d diffusion
Jie Yan Introduction to Probability
![Page 4: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/4.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Contents
I Discrete distribution.
I Continuous distribution.
I Important distributions
I Application: 1-d diffusion
Jie Yan Introduction to Probability
![Page 5: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/5.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Contents
I Discrete distribution.
I Continuous distribution.
I Important distributions
I Application: 1-d diffusion
Jie Yan Introduction to Probability
![Page 6: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/6.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Definition
Consider M discrete events x = xi , i = 1,2, · · · ,M. The probabilityfor the occurrence of xi is:
P(xi ) =Ni
N, (N →∞),
where Ni is the number of occurrence of xi , and N =M∑i=1
is the
total occurrence of all the events. P(xi ) has the following
properties: P(xi ) ≥ 0,M∑i=1
P(xi ) = 1.
Jie Yan Introduction to Probability
![Page 7: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/7.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Mean, standard deviation, and moments
I Mean: < f (xi ) >=M∑i=1
f (xi )P(xi )
I 1st moment: < x >=M∑i=1
xiP(xi )
I 2st moment: < x2 >=M∑i=1
x2i P(xi )
I ......
I nth moment: < xn >=M∑i=1
xni P(xi )
Jie Yan Introduction to Probability
![Page 8: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/8.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Definition
For a continuous variable x ∈ [a, b], we can define the “density ofthe occurrence” n(x), so that n(x)∆x denotes the “number ofoccurrence” in a small interval [x , x + ∆x ]. In the limit ∆x → 0,
the total occurrence is N =b∫a
dxn(x). We then define the
probability density function ρ(x) = n(x)N . ρ(x)∆x = n(x)∆x
N is theprobability of x falling into the small interval [x , x + ∆x ]:P(x ∈ [x , x + ∆x ]). When the interval is not small,
P(x ∈ [x , x + ∆x ]) =∫ x+∆xx dxρ(x). Obviously we have
b∫a
ρ(x) = 1.
Jie Yan Introduction to Probability
![Page 9: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/9.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Mean, variance, moments
I Mean: < f (x) >=b∫a
f (x)ρ(x)
I 1st moment: < x >=b∫a
dxxρ(x)
I 2nd moment: < x2 >=b∫a
dxx2ρ(x)
I ......
I nth moment: < xn >=b∫a
dxxnρ(x)
Jie Yan Introduction to Probability
![Page 10: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/10.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Joint probability distribution
For two sets of events: x = xi , i = 1, 2, · · · ; y = yj , j = 1, 2, · · · .The combined events form a set: (x , y) = (xi , yk), i , j = 1, 2, · · · .The probability of the combined event (xi , yk) is denoted byP(xi , yk) and is called ”the joint probability”. For continuousdistribution, the corresponding term is the joint probability densityfunction ρ(x , y). The joint prob is normalized. If x and y areindependent, we have the following:
I 1. P(xi , yk) = P(xi )P(yk);ρ(x , y) = ρ(x)ρ(y).
I 2. < xy >=< x >< y >;< ((x + y)− < (x + y) >)2 >=<(x− < x >)2 > + < (y− < y >)2 >.
I 3. cor(x , y) =∫ ∫
dxdy(x− < x >)(y− < y >)ρ(x , y) = 0.
Jie Yan Introduction to Probability
![Page 11: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/11.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Addition and multiplication rules
I addition rule applies to exclusive events: for discretedistribution, P(either xi or xj)= P(xi ) + P(xj). Forcontinuous distribution, P(either x ∈ [a, b] or x ∈ [c , d ]) =∫ ba dxρ(x) +
∫ dc dxρ(x). ([a, b] and [c , d ] don’t overlap).
I multiplication rule applies to independent events x and y : fordiscrete distribution,P(xi , yj) = P(xi )P(xj). For continuousdistribution, ρ(x , y) = ρ(x)ρ(y).
Jie Yan Introduction to Probability
![Page 12: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/12.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Binomial distribution
Experiment of throwing a coin: each throw has a fixed probabilityp of ”face up”, and 1− p of ”face down”. We throw N timesindependently (that means the previous throw does not affect thenext throw), the probability to have n times of finding the coin tobe ”face up” is easily derived to be:
P(n) =N!
n!(N − n)!pn(1− p)N−n,
where pn(1− p)N−n is the probability to have n specified coin”face up”. N!
n!(N−n)! is the number of ways we can specify n coinsout from the total number N.
Jie Yan Introduction to Probability
![Page 13: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/13.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Binomial distribution
Memorize: Binomial theorem (a + b)N =∑N
k=0N!
k!(N−k)!akbN−k .
I Mean: prove the binomial distribution isnormalized:
∑Ni=1 P(n) = 1 (hint: binomial theorem).
I 1st moment: memorize < n >= pN (hint: partial derivativewith respect of p).
I 2nd moment: < n2 >= (pN)2 + Np(1− p)
I variance: memorizeσN =< (n− < n >)2 >=< n2 > − < n >2= Npq
Jie Yan Introduction to Probability
![Page 14: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/14.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Gaussian distribution
ρ(x) =1
σ√
2πe−(x−µ)2
2σ2 ,
where x ∈ [−∞,∞]. Please memorize the famous Gaussianintegral formula I (b) =
∫∞−∞ dye−by2
=√
πb , where b ≥ 0. Please
show that dI (b)db = −
∫∞−∞ dyy2e−by2
.Please prove the following exist:∞∫−∞
dxρ(x) = 1; < x >= µ;
Please compute the following: < x2 >; < (x− < x >)2 >; and< x2 > − < x >2. Please show that< (x− < x >)2 >=< x2 > − < x >2.
Jie Yan Introduction to Probability
![Page 15: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/15.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
More on Gaussian distribution
I It is a good approximation to the Binomial distribution atlarge N and Np.
I In statistical physics, ρ(x) ∝ e−E(x)kBT . In many cases we are
dealing with harmonic potential E (x) = 12kx2.
I Central limit theorem: x̄N =∑N
i=1xiN itself satisfies a Gaussian
distribution with µ =< x >, and σ = σx/√
N. The centrallimit theorem says that data which are influenced by manysmall and unrelated random effects are approximatelynormally distributed.
Jie Yan Introduction to Probability
![Page 16: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/16.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
More on Gaussian distribution
I It is a good approximation to the Binomial distribution atlarge N and Np.
I In statistical physics, ρ(x) ∝ e−E(x)kBT . In many cases we are
dealing with harmonic potential E (x) = 12kx2.
I Central limit theorem: x̄N =∑N
i=1xiN itself satisfies a Gaussian
distribution with µ =< x >, and σ = σx/√
N. The centrallimit theorem says that data which are influenced by manysmall and unrelated random effects are approximatelynormally distributed.
Jie Yan Introduction to Probability
![Page 17: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/17.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
More on Gaussian distribution
I It is a good approximation to the Binomial distribution atlarge N and Np.
I In statistical physics, ρ(x) ∝ e−E(x)kBT . In many cases we are
dealing with harmonic potential E (x) = 12kx2.
I Central limit theorem: x̄N =∑N
i=1xiN itself satisfies a Gaussian
distribution with µ =< x >, and σ = σx/√
N. The centrallimit theorem says that data which are influenced by manysmall and unrelated random effects are approximatelynormally distributed.
Jie Yan Introduction to Probability
![Page 18: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/18.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Binomial-Gaussian Approx
Jie Yan Introduction to Probability
![Page 19: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/19.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Application: 1-d random walk
A particle move along a line with a step size b for each movement.The probabilities of move left or to right are equal (p=0.5). Thenet displacement from its original position after N steps (we canassume a large N) is: s = (nr − nl)b. What is the distributionfunction of s?
Jie Yan Introduction to Probability
![Page 20: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/20.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
1-d random walk
Jie Yan Introduction to Probability
![Page 21: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/21.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
1-d random walk: diffusion
Jie Yan Introduction to Probability
![Page 22: Introduction to ProbabilityBiophysics/PC2267/Lecture-02-prob.pdfOutline Discrete distribution Continuous distribution Important distributions Diffusion in 1-d Introduction to Probability](https://reader036.vdocuments.site/reader036/viewer/2022063000/5f0f22277e708231d442a77b/html5/thumbnails/22.jpg)
OutlineDiscrete distribution
Continuous distributionImportant distributions
Diffusion in 1-d
Diffusion in 2-d
Jie Yan Introduction to Probability