Elements of Set Theory
A set S is a collection of elements
i:
[nln4]
S = cfw_ 1 , 2 , . . ..
All elements of subset A are also elements of S : A S .
The empty set contains no elements.
If S contains n elements then the number of subsets is 2n .
Hierarchy
Exponential distribution
[nln10]
Busses arrive randomly at a bus station.
The average interval between successive bus arrivals is .
f (t)dt: probability that the interval is between t and t + dt.
P0 (t) =
dt f (t ): probability that the interval is larger
Waiting Time Problem
[nln11]
Busses arrive more or less randomly at a bus station.
Given is the probability distribution f (t) for intervals between bus arrivals.
Normalization:
dt f (t) = 1.
0
Probability that the interval is larger than t: P0 (t) =
dt f
Central Limit Theorem
[nln9]
The central limit theorem is a major extension of the law of large numbers.
It explains the unique role of the Gaussian distribution in statistical physics.
Given are a large number of statistically independent random variable
Binomial, Poisson, and Gaussian Distributions
Consider a set of N independent experiments, each having two possible outcomes occurring with given probabilities.
events
A+B =S
probabilities
p+q =1
random variables n + m = N
Binomial distribution:
PN (n) =
Statistical Uncertainty and Information
[nln5]
An experiment has n possible outcomes that occur with probabilities P1 , P2 , . . . , Pn .
Properties that must be satised by any quantitative measure of uncertainty:
1. The uncertainty is a function of the p
Multivariate Distributions
[nln7]
Let X = (X1 , . . . , Xn ) be a random vector variable with n components.
Joint probability distribution: P (x1 , . . . , xn ).
Marginal probability distribution:
dxm+1 dxn P (x1 , . . . , xn ).
P (x1 , . . . , xm ) =
Con
Contraction memory time scales
[nln15]
microscopic dynamics
contraction
future state determined
by present state alone
focus on subset of
future state determined
dynamical variables by present and past states
deterministic time evolution
of dynamic vari
[nex4] Educated guess
A railroad company numbers its locomotives in order, 1, 2, . . . , N .
(a) One day, you see a locomotive, and its number is 60. What is your best guess for the total
number N of locomotives which the company owns?
(b) On the followin
Master Equation with detailed balance
[nln12]
Master equation with time-independent transition rates W (n|m) = Wmn :
P (n, t) =
t
[Wmn P (m, t) Wnm P (n, t)] =
Lmn P (m, t),
m
m
where Lmn = Wmn mn
n
Wnn = Wmn mn .
This set of linear, ordinary dierential e