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Understanding Random Variables in Probability Theory

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13 diapositives · 324 Views · Sep 08, 2014

This lecture examines singular, discrete, and continuous random variables, presenting the Strong Law of Large Numbers and the construction of singular Cumulative Distribution Functions using Devil's staircase. Theoretical concepts are illustrated with probabilistic games and examples. Implications of the Strong Law are discussed, contrasting with the Weak Law in the context of independently distributed random variables.

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  1. Lecture 26 Singular random variable Strong law of large number

  2. Discrete Continuous F(x) F(x) 1 1 x x Derivative is zero almost everywhere. Number of discontinuity points are finite or countably infinite. Continuous Random variable Both discrete and continuous r.v. can be described by cumulative distribution function F(x)=P(Xx).

  3. Singular Random variable Singular -- Neither discrete nor continuous. Devil’s staircase: a continuous function rising from 0 to 1 on the unit interval whose derivative is zero almost everywhere. Closely related to the Cantor’s set. Uncountably many steps

  4. Iterative construction % ERG2040A Lecture26 Construction of a singular cdf x = [0 1/3 2/3 1]; y = [0 1/2 1/2 1]; for k=1:6 figure(k) plot(x,y) axis square x = [x/3 2/3+x/3]; y = [y/2 1/2+y/2]; end figure(k+1) plot(x,y) title('Cumulative distribution function of a singular random variable') axis square xlabel('x'); ylabel('F(x)') The Devil’s staircase can be constructed by the following algorithm.

  5. Discrete, continuous and singular • Continuous: cdf is differentiable. • Discrete: Derivative of cdf is zero except for finitely or countably many points. • Singular: Derivative of cdf is zero almost everywhere. Number of non-differentiable points many be uncountably infinitely. • A unifying framework for discrete, continuous and singular is the measure theory and Lebesgue integral (MAT4050/5011/5012.)

  6. “Probability = 0”  Impossible • In the game of CLi vs KKLeung, there are sample paths that are non-terminating. …… However, the event of all such non-terminating paths has probability zero. With probability 1, somebody will win eventually.

  7. 1/4 N 3/4 3/4 1/4 W E 1/4 3/4 3/4 S 1/4 Ma Jong 4 people play Ma Jong for infinitely many times. • P(E wins indefinitely)=0 • P(EEEEEEEEE……..)=0 • P(S wins finitely many times)=0 • P(No body wins two times consecutively)=0, • P(ESWN…, or SWNE…, or WNES…, or NESW…)=0 • P(No body wins more than 10 times consecutively) =0 • P(When ever W loses, N will win in the next game)=0 • P(complement of the above)=1

  8. Normal number theorem • There are (in fact uncountably many) infinitely long 0-1 sequences (X1,X2,X3,…) such that the limit does not exist. • e.g.*11*11*11*11*11*11*11*11…, where * is either 0 or 1) • If the Xi are chosen independently and equally likely, the set of all such sequences has probability zero. Indeed, with probability 1, the limit exists and equals to 0.5.

  9. Strong law Version 1 • Let X1, X2, X3, X4, X5,…, be IID (independent and identically distributed) random variables with finite mean. • Let Sn = X1+ X2 +…+ Xn. • With probability one, • Implications: with probability zero: • the limit does not exists. • the limit exists but Sn/n approaches the wrong limit.

  10. Strong law version 2 “sup” stands for supremum. Roughly speaking, supremum is the maximum. For any >0, In words, if we plot Sn/n against n, with probability 1, The graph is bounded between E[x] eventually, for any >0. E[X]+ E[X] E[X]–

  11. Strong vs Weak No supremum in the weak law Weak Law: for any >0, • Since both strong law and weak law hold in the case of IID random variable, we cannot illustrate their difference using IID r.v’s. • Example: Let Sn, n1, be independent random variables. Sn=n with probability 1/n, and 0 otherwise. • E[Sn]=1 for all n. • We can interpret Sn as sum of X1+…+Xn. Then E[Xn]=0 for all n.

  12. Sn/n Strong law fails in this example • Sn/n is either 0 or 1. • Take =0.5. Fix any n, the event that Sm/m is within 00.5 for all mn has probability which can be shown equal to zero. • But weak law holds. 1 n

  13. H20 • Close book, open notes with no page limitation. Bring calculator • 6 questions. • No proof, no bonus. • Include material before and during midterm. • P(Combinatorics)=1. • P(probability Density function, cumulative distribution function) = 1 • P(Mean, variance, covariance) = 1 • P(Gaussian, exponential)=1 • P(Central limit theorem)=1 • P(Chebyshev inequality)=1 • P(Markov chain)=1 • P(Cauchy-Schwarz inequality)=0.5 • P(Characteristic function)=0 • P(Content in this file)=0