Department of Mathematics Ma 3/103 KC Border Introduction to Probability and Statistics Winter 2021 Lecture 7: Concentration Inequalities and the Law of Averages Relevant textbook passages: Pitman [19]: Section 3.3 Larsen–Marx [17]: Section 4.3 7.1 The Law of Averages The “Law of Averages” is an informal term used to describe a number of mathematical theorems that relate averages of sample values to expectations of random variables. F ∈ Given random variables X1,...,Xn on a probability space (Ω, ,PP), each point ω Ω yields ¯ n a list X(ω)1,...,X(ω)n. If we average these numbers we get X(ω) = i=1 Xi(ω)/n, the sample average associated with the outcome ω. The sample average, since it depends on ω, is also a random variable. In later lectures, I’ll talk about how to determine the distribution of a sample average, but we already have a case that we can deal with. If X1,...,Xn are independent Bernoulli random variables, their sum has a Binomial distribution, so the distribution of the sample average is easily given. First note that the sample average can only take on the values k/n, for k = 0, . , n, and that n P X¯ = k/n = pk(1 − p)n−k. k Figure 7.1 shows the probability mass function of X¯ for the case p = 1/2 with various values of n. Observe the following things about the graphs. • The sample average X¯ is always between 0 and 1, and it is simply the fraction of successes in sequence of trials. • If the frequency interpretation of probability is to make sense, then as the sample size grows, it should converge to the probability of success, which in this case is 1/2. • What can we conclude about the probability that X¯ is near 1/2? As the sample size becomes larger, the heights (which measure probability) of the dots shrink, but there are more and more of them close to 1/2. Which effect wins? What happens for other kinds of random variables? Fortunately we do not need to know the details of the distribution to prove a Law of Averages. But we start with some preliminaries. KC Border v. 2020.10.21::10.28 Ma 3/103 Winter 2021 KC Border The Law of Averages 7–2 ��� ��� ��� ������� �� �� ����������� ��������� ��� 0.20 ● ● ● 0.15 ● ● 0.10 ● ● 0.05 ● ● ● ● ● ● ● ● ● ● 0.2 0.4 0.6 0.8 1.0 ��� ��� ��� ������� �� ��� ����������� ��������� ��� 0.05 ●●● ● ● ● ● ● ● ● ● 0.04 ● ● ● ● 0.03 ● ● ● ● ● ● 0.02 ● ● ● ● ● ● ● ● 0.01 ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.2 0.4 0.6 0.8 1.0 ��� ��� ��� ������� �� ������ ����������� ��������� ��� ● ● ● 0.0030 ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● 0.0025 ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 0.0020 ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 0.0015 ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 0.0010 ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 0.0005 ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 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0.2 0.4 0.6 0.8 1.0 Figure 7.1. v. 2020.10.21::10.28 KC Border Ma 3/103 Winter 2021 KC Border The Law of Averages 7–3 7.2 Standardized random variables Pitman [19]: p. 190 7.2.1 Definition Given a random variable X with finite mean µ and finite variance σ2, the standardization of X is the random variable X∗ defined by X − µ X∗ = . σ Note that E X∗ = 0, and Var X∗ = 1, and X = σX∗ + µ, so that X∗ is just X measured in different units, called standard units. ∗ [Note: Pitman uses both X and later X∗ to denote the standardization of X.] A convenient feature of standardized random variables is that they are invariant under change of scale and location. 7.2.2 Proposition Let X be a random variable with mean µ and standard deviation σ, and let Y = aX + b, where a > 0. Then X∗ = Y ∗. Proof : The proof follows from Propositions 6.7.1 and 6.10.2, which assert that E Y = aµ + b and SD Y = aσ. So z =}|Y { Y − aµ − b aX + b −aµ − b a(X − µ) X − µ Y ∗ = = = = = X∗. aσ aσ aσ σ 7.3 Concentration inequalities The term concentration inequality refers to a proposition about the probability of the value of a random variable being concentrated near a particular point. Concentration inequalities are crucial to the proof of the Law of Large Numbers and have many applications in statistics. One could write a whole book on the subject. Indeed Boucheron, Lugosi, and Massart [2] have done so. 7.3.1 Markov Markov’s Inequality bounds the probability of large values of a nonnegative random variable in terms of its expectation. It is a very crude bound, but it is just what we need for the Weak Law of Large Numbers. Pitman [19]: p. 174 7.3.1 Proposition (Markov’s Inequality) Let X be a nonnegative random variable with finite mean µ. For every ε > 0, µ P (X > ε) 6 . ε KC Border v. 2020.10.21::10.28