introduction to probabilityphyweb.physics.nus.edu.sg/~biophysics/pc2267/lecture-02... · 2006. 8....

OutlineDiscrete distribution

Continuous distributionImportant distributions

Diffusion in 1-d

Introduction to Probability

Jie Yan

August 18, 2006

Jie Yan Introduction to Probability



Diffusion in 1-d

Contents

I Discrete distribution.

I Continuous distribution.

I Important distributions

I Application: 1-d diffusion




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.




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 )




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.




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)




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.




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).




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.




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




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.




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.




Diffusion in 1-d

Binomial-Gaussian Approx




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?




Diffusion in 1-d

1-d random walk




Diffusion in 1-d

1-d random walk: diffusion




Diffusion in 1-d

Diffusion in 2-d


introduction to probabilityphyweb.physics.nus.edu.sg/~biophysics/pc2267/lecture-02... · 2006. 8....

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