cf. The Fields medal carries a portrait of Archimedes.

Transcription

1 Review Gas kinetics cannot give us any info about atoms. [how about QM?] 2.1: I forgot to stress the crucial importance of Archimedes ( BCE). cf. Archimedes.pdf cf. The Fields medal carries a portrait of Archimedes. The Hellenistic tradition was totally lost due to Christian bigotry and Völkerwanderung (Barbarian Invasions). Cauchy ( ) at last caught up with Archimedes. cf m and m. 1

2 Probability measure: What is probability? What is measure? Probability = normalized measure Probability theory is measure theory with a soul. (M Kac) ω: Probability parameter specifying elements in Ω. Probability theory does not care whether a probability under study is reasonable or not. Natural selection has molded our emotion/intuition. subjective probability rational probability However, probability estimate for social events may well be warped. 2

3 Lebesgue integral an advertisement a b Simple functions: functions with finitely many values n f(x) = c i χ {x [a,b]:f(x)=ci }(x). Then, b a i=1 dx f(x) = c i total length of {x [a, b] : f(x) = c i } i When we compute the integral of a function F, it is approximated by a simple function sequence (cf. the above figure). 3

4 Dirichlet s function D(x) is a indicator of Z: { 1 if x Z, D(x) = 0 otherwise. Integrate this function on [0, 1]. Riemann cannot compute this. 1 0 D(x)dx = 1 Lebesgue measure of [0, 1] Z Since Z is countable (denumerable), [0, 1] Z is measure zero: m ([0, 1] Z) = 0. end 4

5 Lecture 3. Law of Large Numbers Intuitive and elementary LLN Why plausible? The variation of the empirical average is O[1/N]. [demo] Elementary LLN [demo: Mathematica & R] V N = ( ) 2 ( 1 1 dp N S N m > dp N 1 S N m >ɛ N S N m ( ) > ɛ 2 dp = ɛ 2 1 P N S N m > ɛ. 1 N S N m >ɛ ) 2 How to observe probability: 1 N χ A (x i ) P (A). N i=1 Here, x i is the i-th observation result. Relation to frequency is derived! 5

6 Why is the elementary version weak? (1) It is for an ensemble, not for individual samples. (2) Very large fluctuations can still happen for large N: S N /N m > ɛ with probability of O[1/N] Strong law: cf. LLN review For any ɛ > 0, there is N such that for any n > N S N /N m < ɛ for almost all runs. We should distinguish various convergences in probability theory. 6

7 Convergences in Probability Theory The strongest convergence is: Almost sure convergence: X n (ω) X(ω) for ω Ω. Convergence in probability: for any ɛ > 0 there is N such that P ( X n X > ɛ) = 0. Convergence in mean (L p -convergence): E( X n X p ) 0. Convergence in distribution: The distribution F n of X n converges to F of X. almost sure in probability in distribution in mean 7

8 Notable theorems Proofs may be posted [Kolmogorov s 01 law] Let {X n } be independent stochastic variables. If event A depends infinitely many of these variables, P (A) = 0 or 1. cf. Thermodynamics [Regellosigkeit] Let X k be independent of {X 1,, X k 1, Y 1,, Y k } for k = 1,, n, E(X k ) = m, V (X k ) v, P (Y k [0, 1]) = 1 and P ( Y k = ) = 1. Then, N Yn X n / N Yn m (a.e.), e.g., however you choose samples, if infinitely many, there is no difference. cf. Gambler s fallacy 8

9 Strong Law demonstration See LLNReview.pdf (self-contained) Proof strategy: (1) We may assume X n 0. (2) We may assume X n < n for all n. (3) Use the weak law and Borel-Cantelli Lemma. I do not demand all of you to understand math details, but hope all of you to feel theorems very plausible. 9

10 (1) X ± n = max(±x n, 0): X n = X + n X n, X n = X + n + X n. We may separately consider X ± n. (2) X n could be very large, but X n > n should not occur very often. Actually for any sample run, this is almost surely finite. We can show P (X n > n) < E(X 1 ) <. n=1 Then, the Borel-Cantelli lemma tells us X n > n cannot happen infinitely many times. This requires P (X n > n) > 0, n=m but implies the contrary. 10

11 We wish to show 1 N N n=1 X n converges for almost all ω. (3) Kronecker tells us a sufficient condition is the convergence of N 1 n X n. n=1 (4) This is guaranteed by Kolmogorov s inequality (stronger Chebyshev): m P ( (x k /k) > λ) 1 m 1 λ 2 k k). 2E(X2 k=n k=n and (5) convergence of the RHS for m. 11

12 12

13 (Implicit) applications of (strong) law of large numbers Internal energy is (almost) constant (4.5): T is well defined. Number of particles in V/2 is N/2. Number density n(t) = 1 dτ N χ dτ (r i (t)), i=1 with a uniform distribution of the particle position: E(χ dτ (r i (t))) = dτ/v. V dτ 13

14 Maxwell s distribution 5.1: Probability density distribution: f(x) = P (dx) dx (Radon-Nikodym derivative). 5.2 Maxwell distribution via Cauchy s functional equation How to compute Gaussian integrals 5.7 Boltzmann factor e βu. (1) How can we empirically verify Maxwell s distribution? (2) Quantum mechanics? 14