Probability & Statistics

1. Probability & Statistics Brief Overview of Statistical Methods (at most, 2 classes) Note: Many more detailson the probability topics discussed here may be found on the Lecture page for my Physics 4302 course: http://www.phys.ttu.edu/~cmyles/Phys4302/lectures_Ch1.html

2. Brief discussion of the pure • Math of Probability • & Statistics. “The true logic of this world is in the calculus of probabilities” James Clerk Maxwell

3. Math of Probability & Statistics. “Misunderstanding of probability may be the greatest of all impediments to scientific literacy.” Stephen Jay Gould

4. Relevance of Probability to Physics • In this course, we’ll discuss The Physicsof systems containing HUGE numbers ( >> 1023) of particles:  Solids, Liquids, Gases, EM Radiation,… (photons & other quantum particles), • Challenge:Describe a system’s Macroscopic characteristics starting from a Microscopictheory.

5. GOAL: Formulate a theory to describe a system’s Macroscopic characteristics starting from a Microscopictheory. • Classical Mechanics: Newton’s Laws Need to solve >>1023coupled differential Newton’s 2nd Law equations of motion! (ABSURD!!) • Quantum Mechanics:Schrödinger’s Equation: Need a solution for >>1023 particles! (ABSURD!!)

6. Historically, this led to the use of a Statistical description of such a system. So, we’ll talk about Probabilities & Average System Properties • We AREN’Tconcerned with detailed behavior of individual particles.

7. Definitions: • Microscopic:~ Atomic sized; ~ ≤ a fewÅ • Macroscopic:Large enough to be “visible” in the “ordinary” sense • An Isolated Systemis in Equilibriumwhen it’s Macroscopic parameters are time-independent. The usual case in this course! • But, note! Even if it’s Macroscopic parameters are time-independent, a system’s Microscopic parameters can & probably will still vary with time!

8. Now, some BasicMath of Probability & Statistics “The most important questions of life are, for the most part, really only problems of probability.” Pierre Simon Laplace “ThéorieAnalytique des Probabilités”, 1812

9. The Binomial Probability Distribution • The following should hopefully be a review! (?) • Keep in mind: Whenever we want to describe a situation using probability & statistics, we must consider an assembly of a large number N(in principle, N ∞) of “similarly prepared systems”. This assembly is called an ENSEMBLE (“Ensemble” = French word for Assembly). • The Probability of an occurrence of a particular event is DEFINED with respect to this particular ensemble & is given by the fraction of systems in the ensemble characterized by the occurrence of this event.

10. Example: In throwing a pair of dice, give a statistical description by considering that a very large number Nof similar pairs of dice are thrown under similar circumstances. • Alternatively, we could imagine the same pair of dice thrown N times under similar circumstances. The probability of obtaining two 1’s is then given by the fraction of these experiments in which two 1’s is the outcome. • Note that this probability depends stronglyon the type of ensemble to which we are referring.

11. To quantitatively introduce probability concepts, we use a specific, simple example, which is much more generalthan you might think. This is: The Binomial Probability Distribution • This is illustrated in Ch. 1 of the supplemental book by Reif, where he discusses the “1 Dimensional Random Walk Problem”

12. “1 Dimensional Random Walk Problem” • This is also discussed in a HUGE numberof other books & in many places on the web. In the following, details of derivations will, for the most part, not be shown. Instead, the results will be summarized. See other sources for derivation details

13. The One-Dimensional Random Walk • In it’s simplest, crudest, most idealized form, The random walk problem can be viewed as in the figure • A story about this is that a drunk that starts out from a lamp post on a street. Obviously, he wants to move down the sidewalk to get somewhere!!

14. So the drunk starts out from a lamp post on a street. Each step he takes is of equal lengthℓ. He is SO DRUNK, that the direction of each step (right or left) is completely independent of the preceding step. • The (assumed known) probability of stepping to the right is p & of stepping to the left is q = 1 – p. In general, q ≠ p. • The x axis is along the sidewalk, the lamp post is at x = 0. Each step is of length ℓ, so his location on the x axis must be x = mℓ where m = a positive or negative integer.

15. Question: After N steps, what is the probability that the man is at a specific location x = mℓ (m specified)? • To answer, we must first consider an ensemble of a large numberNof drunk men starting from similar lamp posts!! • Or repeat this with the same drunk man walking on the sidewalk N times!!

16. This is “easily generalized” to 2 dimensions, as shown schematically in the figure. • The 2 dimensional random walk corresponds to a PHYSICS problem of adding N, 2 dimensional vectorsof equal length (figure) & random directions & asking: “What is the probability that the resultant has a certain magnitude & a certain direction?”

17. Physical Examplesto which the Random Walk Problemapplies: 1. Magnetism(Quantum Treatment) • Natoms, each with magnetic moment μ. Each has spin ½. By Quantum Mechanics, each magnetic moment can point either “up” or “down”. If these are equally likely, what is the Net magnetic moment of the N atoms? 2. Diffusion of a Molecule of Gas(Classical) • A molecule travels in 3 dimensions with a mean distance ℓbetween collisions. How far is it likely to have traveled after Ncollisions? Answer using Classical Mechanics.

18. Random Walk Problem: • Is a simple example which illustrates some fundamental results of Probability Theory. • The techniques used are Powerful & General. • They are used repeatedly throughout Statistical Mechanics. • So, it’s important to spend a bit of time on this problem & to understand it!

19. 1-Dimensional Random Walk • Forget the drunk! Go back to Physics! • Think of a particle moving in 1 dimension in steps of length ℓ, with 1. Probability p of stepping to right & 2. Probabilty q = 1 – p of stepping to left. • After N steps, the particle is at position: x = mℓ (m = integer; - N ≤m≤N). Let n1≡ # of steps to the right (out of N) Let n2≡ # of steps to the left.

20. After N steps, x = mℓ (- N ≤m≤N). • Let n1≡ # of steps to the right (out of N) • Let n2≡ # of steps to the left. • Clearly, N = n1+ n2(1) • Clearly also, x ≡ mℓ = (n1- n2)ℓ or, m = n1- n2(2) • Combining (1) & (2) gives:  m = 2n1– N (3) So, if N is odd, so is m & if N is even, so is m.

21. A Fundamental Assumption is that Successive Steps are Statistically Independent • Let p≡ the probability of stepping to the right & q = 1 – p≡ the probability of stepping to the left. (p + q = 1) • Since each step is statistically independent, the probability of a given sequence of n1steps to the right followed by n2steps to the left is given by multiplying the respective probabilities for each step. (N = n1 + n2)

22. A Detailed Derivation of the Probability Distribution Gives: • The probability WN(n1) of taking N steps; n1 to the right & n2 (=N -n1) to the left is WN(n1) = [N!/(n1!n2!)]pn1qn2or WN(n1) = [N!/{n1!(N – n1)!]}pn1(1-p)n2 • Often, this is written as WN(n1)  N pn1qn2 n1 Remember that q = 1- p

23. WN(n1) = [N!/{n1!(N – n1)!]}pn1(1-p)n2 • This probability distribution is called the Binomial Distribution. • This is because the Binomial Expansion has the form (p + q)N = ∑(n1 = 0N)[N!/[n!(N–n1)!]pn1qn2 • Change variables & ask for the probabilityPN(m) that x = mℓafter N steps. PN(m) = WN(n1).But m = 2n1– N, so n1= (½)(N + m) & n2= N - n1 = (½)(N - m). Results on next slide:

24. PN(m) = {N!/([(½)(N + m)]![(½)(N – m)!]}  p(½)(N+m)(1-p)(½)(N-m) • For the common case of p = q = ½, this is: PN(m) = {N!∕([(½)(N + m)]![(½)(N – m)!])}(½)N • This is the usual form of the Binomial Distribution which is probably the most elementary (discrete) probability distribution.

25. As a trivial example, suppose that p = q = ½, N = 3 steps: • This gives, P3(m) = {3!/[(½)(3+m)!][(½)(3-m)!](½)3 • SoP3(3) = P3(-3) = (3!/[3!0!](⅛) = ⅛ P3(1) = P3(-1) = (3!/[2!1!](⅛) = ⅜ Table of Possible Step Sequences

26. Another example, let: p = q = ½, N = 20. • This gives: P20(m) = {20!/[(½)(20 + m)!][(½)(20 - m)!](½)3 • Calculation gives the histogram results in the figure. P20(20) = [20!/(20!0!)](½)20 P20(20)  9.5  10-7 P20(0) = [20!/(10!)2](½)20 P20(0)  1.8  10-1

27. Notice: The “envelope” of the histogram is a “bell-shaped” curve. The significance of this is that, after N random steps, the probability of a particle being a distance of N steps away from the start beomes very small & the probability of it being at or near the origin becomes relatively large: • For this same case: P20(m) = {20!/[(½)(20 + m)!][(½)(20 - m)!](½)3 P20(20) = [20!/(20!0!)](½)20 P20(20)  9.5  10-7 P20(0) = [20!/(10!)2](½)20 P20(0)  1.8  10-1