CSSS 2000–2001 Math Review Lectures: Probability ...

1 CSSS 2000 2001 Math Review Lectures: Probability , statistics and stochastic ProcessesCosma Rohilla ShaliziUniversity of Wisconsin-Madison, Physics Department, and the Santa Fe 4 June 2000and 11 June 20012iThese notes are available electronically shalizi/prob-notes/.Reports of typos, and more serious bugs, are eagerly HistoryDateComments4 June 2000 First version finished (about four hours before teaching from it).8 July 2000 Minor typo fixes, some points amplified in response to student comments;added list of deficiencies as Appendix June 2001 More typo fixes, added paragraph on large deviation principle, second July 2001 Fixed two errors (thanks to Cecile Viboud and Antonio Ramirez)ContentsPart I: Probability11 What s a Probability Anyway?12 Probability Basic Rules .. Caution about Probabilities 0 and 1 .. Conditional Probabilities ..33 Random Properties of Random Variables .. of Random Variables .. Random Variables; Independence .. Expectation.

2 Of Multiple Variables .. Moments .. Important Moments ..8 Mean ..8 Variance and Standard Deviation ..84 Important Discrete The Bernoulli Distribution .. The Binomial Distribution .. The Poisson Distribution .. First Moments of These Distributions .. 105 Continuous Random The Cumulative Distribution Function .. The Probability Density Function .. Continuous Expectations .. 126 Important Continuous The Exponential Distribution .. The Normal or Gaussian Distribution .. The 2 Distribution .. The Lognormal Distribution .. Power-Law Distributions .. First Moments and pdfs of These Distributions .. 157 Tricks with Random The Law of Large Numbers .. The Central Limit Theorem .. Extraordinary Importance of the CLT .. in Which the CLT Can Fail .. Many-Independent-Causes Story for Why Things AreGaussian .. Noise and the Lognormal Distribution.

3 19 Part II: Statistics208 The Care and Handling of Counting .. Question of Bins .. Adding .. Mean .. Variance .. Correlating .. Correlation Coefficient .. 239 The Notion of A Statistic .. Loss of Variability Under Sampling .. Figuring the Sample Distribution; Monte Carlo Methods .. 2610 Point Estimates .. Bias, Variance, and Other Sorts of Quality .. Some Common Kinds of Estimates .. 28 Least Squares .. 28 Maximum Likelihood .. Curve-Fitting or Regression .. Propagation of Errors .. Confidence Regions .. 31ivCONTENTS11 Hypothesis Goodness-of-Fit .. Significant Lack of Fit .. The 2 Test .. The Null Hypothesis and Its Rivals .. TheStatus QuoNull .. The It-Would-Be-the-Worst-Mistake Null .. The Random-Effects Null .. The Alternative Hypothesis .. Formalism of Tests .. The Test Statistic.

4 The Regions .. The Kinds of Errors; Error Probabilities .. 37 Significance Level or Size .. 37 Power .. 38 Severity .. 38 The Trade-Offs .. Test for Whether Two Sample Means Are Equal .. 3912 Funky Nonparametric Estimation and Fitting .. Machine Learning .. Causal Inference .. Ecological Inference .. Optimal Experimental Design .. 42 Part III: stochastic Processes4313 Sequences of Random Representing stochastic processes with Operators .. Important Properties of stochastic processes .. Stationarity .. Ergodicity .. Mixing .. 4514 Markov Markov Chains and Matrices .. Some Classifications of States, Distributions and Chains .. Higher-Order Markov Chains .. Hidden Markov Models .. 4815 Examples of Markov Bernoulli Trials .. Biased Drift on a Ring .. The Random Walk .. 49 CONTENTSv16 Continuous-Time stochastic The Poisson Process.

5 Uses .. Brownian Motion, or the Wiener Process .. 52A Notes for Further Probability .. statistics .. stochastic processes .. 55B What s Wrong with These Notes57viCONTENTSC hapter 1 What s a ProbabilityAnyway?As far as pure Probability theory is concerned, probabilities are real numbersbetween 0 and 1, attached to sets in some mathematical space, assigned in away which let us prove nifty theorems. We re not going to worry about any ofthese probabilities make good models of the frequencies with whichevents occur, somewhat in the same way that Euclidean geometry makes a prettygood model of actual space. The idea is that we have a space of occurrenceswhich interest us ourprobability space. We carve this up into (generallyoverlapping) sets, which we callevents. Pick out your favorite event A, andkeep track how often occurrences in A happen, as a proportion of the totalnumber of occurrences; this is thefrequencyofA. In an incredibly widerange of circumstances, frequencies come very close to obeying the rules formathematical probabilities, and they generally come closer and closer the longerwe let the system run.

6 So we say that the Probability of A, P(A), is the limitingvalue of the frequency of :probabilities are numbers which tell us how often things foundations of Probability are one of the most acrimoniously disputed topics in math-ematics and natural science; what I m spouting here is pretty orthodox among stochasticprocess people, and follows my own prejudices. The main alternative to the frequentist lineis thinking that probabilities tell you how much you should believe in different notions. This subjectivist position is OK with assigning a Probability to say the proposition that aperpetual motion machine will be constructed within a 2 Probability CalculusThe function which takes an event and gives us its Probability is called theprobability distribution, theprobability measure, or simply thedistri-bution. It generally isn t defined foreverypossible subset of the probabilityspace; the ones for which it is defined, the good (technically: measured)events, are sometimes called the field of the distribution.

7 We won t need thatterminology, but it s good not to be frightened of it when you run across Basic RulesHere A and B are any two events. Following custom, is the special eventwhich contains every point in our Probability space, and is the event whichcontains no points, the empty set. A is thecomplementof A, the event whichis all the occurrences which are not in A. A+B is the union of A and B; AB isthe intersection of A and B. (Both A+B and AB are also events.)1. 0 P(A) 1 (Events range from never happening to always happening)2. P( ) = 1 (Somethingmust happen)3. P( ) = 0 (Nothing never happens)4. P(A) + P( A) = 1 (Amust either happen or not-happen)5. P(A + B) = P(A) + P(B) P(AB)The last rule could use a little elaboration. The meaning of A+B is Aalone, or B alone, or both together . To figure out how often it happens, weadd how often A and B happen (P(A) + P(B)) buteachof those includes Aand B happening together, so we re counting those occurrences twice, and needto subtract P(AB) to get the right follows are some simple exercises which give you useful rules for ma-nipulating probabilities.

8 Some of them should be trivial, but do the CONDITIONAL yourself that P(AB) P(A). yourself that P(A + B) = P(A) + P(B) if A and B aremutually exclusive yourself that if A0,A1,..Anare mutually exclusive andjointly exhaustive, then P(A0) = 1 ni=1P(Ai). (4) and (5) to show (2); or use (2) and (5) to show (4). A Caution about Probabilities 0 and 1 is not necessarily the only event with Probability 1, nor the only one withprobability zero. In general, Probability 0 means that an event happens so rarelythat in the limit we can ignore it, but that doesn t mean 1 events are said to happen almost always , or almost surely ( ) or almost everywhere ( ), while Probability 0 eventshappen almost never . yourself that if there is an event A6= for which P(A) =0, then there are events smaller than with Probability Conditional ProbabilitiesSuppose you re interested only in part of the Probability space, the part whereyou know some event call it A has happened, and you want to know howlikely it is that various other events B for starters have also you want is theconditional probabilityof B given A.

9 We write thisP(B|A), pronouncing the vertical bar|as conditioned on or given . We canwrite this in terms of unconditional probabilities:P(B|A) P(AB)P(A),( )which makes some sense if you stare at it long probabilities are probabilities, and inherit all the necessary prop-erties; just re-write (1) (5) above with bars and extra letters in the right (Get used to seeing the bars.)If P(B|A) = P(B), then whether or not A happens makes no difference towhether B happens. A and B are then said to beindependentorstatisticallyindependent. (If B is independent of A, then A is independent of :show this.) It is often extremely useful to break Probability problems up intostatistically independent chunks, because there s a lot of machinery for provingresults about may be worrying about what happens when P(A) = 0. That is one of the conditionsunder which conditional probabilities can fail to exist but sometimes they re mathematicallywell-defined even when the event we condition on has zero Probability .

10 If youreallywant toworry about this, read Billingsley (1979).4 CHAPTER 2. Probability CALCULUSC onditional probabilities can be inverted. That is,P(A|B) =P(B|A)P(A)P(B).( )This relationship is calledBayes s Rule,after the Rev. Mr. Thomas Bayes(1702 1761), who did not discover Bayes s Rule from the definition of conditional A0,A1,..Anare mutually exclusive and jointly exhaus-tive events. Prove the following form of Bayes s Rule:P(Ai|B) =P(B|Ai)P(Ai) nj=1P(B|Aj)P(Aj)( ) that A and B are independent if and only if P(AB) =P(A)P(B).A and B are said to beconditionally independentgiven C (orindepen-dent conditional onC) when P(AB|C) = P(A|C)P(B|C). A and B are independent, are they still necessarily independentwhen conditioning on any set C?Chapter 3 Random VariablesEssentially anything which has a decent Probability distribution can be a ran-dom variable. A little more formally, any function of a Probability space,f: 7 , is a random variable, and turns its range (the space ) into aprobability space in turn.