Probability Lecture III (August, 2006)

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1 robability Lecture III (August, 2006) 1 Some roperties of Random Vectors and Matrices We generalize univariate notions in this section. Definition 1 Let U = U ij k l, a matrix of random variables. Suppose E U ij < for all i, j. Define the expectation of U by E(U) = (E(U ij )) k l. The following are some properties of random vectors: Let U, respectively V, denote a random k, respectively l, vector. 1. If A m k, B m l are nonrandom and EU, EV are defined, then E(AU + BV) = AE(U) + BE(V). Definition 2 For a random vector U, suppose EUi 2 < for i = 1,, k or equivalently E( U 2 ) <, where denotes Euclidean distance. Define the variance of U, often called the variance-covariance matrix, by Var(U) = E(U E(U))(U E(U)) T = (Cov(U i, U j )) k k a symmetric matrix. 2. If A is m k as before, Var(AU) = A Var(U)A T. Note that Var(U) is k k, Var(AU) is m m. 3. Let c k 1 denote a constant vector. Then Var(U + c) = Var(U). Var(c) = (0) k k. 4. The variance of any random vector is nonnegative definite symmetric matrix. To see this, note that if a k 1 is constant we can apply Var(AU) = A Var(U)A T to obtain Var(a T U) = Var(Σ k j=1a j U j ) = a T Var(U)a = Σ i,j a i a j Cov(U i, U j ). Because the variance of any random variable is nonnegative and a is arbitrary, we conclude from the above equalities that Var(U) is a nonnegative definite symmetric matrix. Definition 3 Define the moment generating function (m.g.f.) of U k 1 for t R k by 5. If U k 1, V k 1 are independent then M(t) = M U (t) = E(e tt U ) = E(e Σk j=1 t ju j ). M U+V (t) = M U (t)m V (t). 1

2 2 The Bivariate Normal Distribution The family of k-variate normal distributions arises on theoretical grounds when we consider the limiting behavior of sums of independent k-vectors of random variables. In this section we focus on the case k = 2 where all properties can be derived relative easily. A planar vector (X, Y ) has a bivariate normal distribution if, and only if, there exist constants a ij, 1 i, j 2, µ 1, µ 2, and independent standard normal random variables Z 1, Z 2 such that X = µ 1 + a 11 Z 1 + a 12 Z 2 Y = µ 2 + a 21 Z 1 + a 22 Z 2. In matrix notation, if A = (a ij ), µ = (µ 1, µ 2 ) T, X = (X, Y ) T, Z = (Z 1, Z 2 ) T, the definition is equivalent to Two important properties follow from the definition. X = AZ + µ. (1) roposition 4 The marginal distributions of the components of a bivariate normal random vector are (univariate) normal or degenerate (concentrate on one point). Note that the converse of the proposition is not true (See problem B.4.10 in Bickel and Doksum [2001]). Also, note that and define E(X) = µ 1 + a 11 E(Z 1 ) + a 12 E(Z 2 ) = µ 1, E(Y ) = µ 2 σ 1 = Var X, σ 2 = Var Y. Then X has N (µ 1, σ 2 1) and Y a N (µ 2, σ 2 2) distribution. roposition 5 If we apply an affine transformation g(x) = Cx + d to a vector X, which has a bivariate normal distribution, then g(x) also has such a distribution. This is clear because CX + d = C(AZ + µ) + d = (CA)Z + (Cµ + d). (2) We define the variance-covariance matrix of (X, Y ) (or of the distribution of (X, Y )) as the matrix of central second moments ( ) σ 2 Σ = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2, (3) where ρ = Cor(X, Y ) = Cov(X, Y ) σ 1 σ 2. This symmetric matrix is in many ways the right generalization of the variance to two dimensions. Theorem 6 Suppose that σ 1 σ 2 0 and ρ < 1. Then the density of X is p X (x) = ( 1 2π det Σ exp 1 ) 2 (x µ)t Σ 1 (x µ). (4) Remark 7 From (3) we see that Σ is nonsigular iff σ 1 σ 2 0 and ρ < 1. Bivariate normal distribution with σ 1 σ 2 0 and ρ < 1 are referred to as nondegenerate, whereas others are degenerate. 2

3 Remark 8 When ρ = 0, p X (x) becomes the joint density of two independent normal variables. Thus, in the bivariate normal case, correlation zero is equivalent to independence. Exercise 9 Given nonnegative constants σ 1, σ 2, a number ρ such that ρ < 1 and numbers µ 1, µ 2, construct a random vector (X, Y ) T, where X = µ 1 + σ 1 Z 1, Y = µ 2 + σ 2 (ρz ρ 2 Z 2 ) Check that (X, Y ) T has a bivariate normal distribution with vector of means (µ 1, µ 2 ) T and variancecovariance matrix Σ as given in (3). 3 Convergence in robability v.s. in Distribution Chebyshev s inequality If X is any random variable and a is a constant, then 3.1 Converagence in robability ( X a) E(X2 ) a 2. Definition 10 If a sequence of random variables, {Z n }, is such that ( Z n α > ε) approaches zero as n approaches infinity, for any ε > 0 and where α is some scalar, then Z n is said to converge in probability to α. Definition 11 A sequence of random vectors Z n (Z n1, Z n2,..., Z nd ) T converages in probability to Z (Z 1, Z 2,..., Z d ) T iff or equivalently Z nj Zj for 1 j d A Law of Large Numbers Z n Z 0 Theorem 12 Example 13 Let X 1, X 2,, X i, be a sequence of independent random variables with E(X i ) = µ and V ar(x i ) = σ 2. Let X n = 1 n n i=1 X i. Then, for any ε > 0, roof. We first find E( X n ) and V ar( X n ): Since the X i are independent, ( Xn µ > ε) 0 as n E( X n ) = 1 n V ar( X n ) = 1 n 2 n E(X i ) = µ i=1 n i=1 V ar(x i ) = σ2 n The desired result now follows immediately from Chebyshev s inequality, which states that ( Xn µ > ε) V ar( X n ) ε 2 = σ2 nε 2 0, as n 3

4 3.2 Convergence in Distribution Definition 14 Let X 1, X 2, be a sequence of random variables with cumulative distribution functions F 1, F 2,, and let X be a random variable with distribution function F. We say that X n converges in distribution to X if lim n F n(x) = F (x) at every point at which F is continuous. Definition 15 A sequence {Z n } of random vvectors converges in law (in distribution) to Z, written Z n Z, iff h(z n ) for all functions h : R d R, h continuous Central Limit Theorem L h(z) Theorem 16 The Multivariate Central Limit Theorem. Let X 1, X 2,, X n be independent and identically distributed random k vectors with E X 1 2 <. Let E(X 1 ) = µ, V ar(x 1 ) = Σ, and let S n = Σ n i=1 X i. Then, for every continuous functiong: g : R k R, ( ) Sn nµ L g g(z) n where Z N k (0, Σ) The O, p, and o Notation The following asymptotic order in probability notation is useful. U n = o (1) iff U n 0 U n = O (1) iff ɛ > 0, M < such that n [ U n M] ɛ U n = o (V n ) iff U n V n = o (1) U n = O (V n ) iff U n V n = O (1) U n p V n iff U n = O (V n ) and V n = O (U n ). Note that O (1)o (1) = o (1), O (1) + o (1) = O (1), and U n L U = Un = O (1). Example 17 Suppose Z 1, Z 2,, Z n are iid as Z 1 with E Z 1 2 <. Set µ =E Z 1, then Z n = µ+o p (n 1 2 ) by the central limit theorem. Exercise 18 Let X i be the last digit of Di 2, where D i is a random digit between 0 and 9. For instance, if D i = 7 then Di 2 = 49 and X i = 9. Let X n = (X X n )/n be the average of a large number n of such last digits, obtained from independent random digits D 1,, D n. 4

5 a) redict the value of Xn for large n. b) Find a number ɛ such that for n = 10, 000 the chance that your prediction is off by more than ɛ is about 1 in 200. c) Find approximately the least value of n such that your prediction of Xn is correct to within 0.01 with probability at least d) Which can be predicted more accurately for large n: the value of Xn, or the value of D n = (D D n )/n? e) If you just had to predict the first digit of X100, what digit should you choose to maximize your chance of being correct, and what is that chance? References [1] eter J. Bickel and Kjell A. Doksum (2001) Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I, 2nd Edition. rentice Hall [2] Geoffrey Grimmett and David Stirzaker (2002) robability and Random rocesses, 3rd Edition, Oxford University ress. [3] Jim itman (1993) robability, Springer-Verlag New York, Inc. [4] John A. Rice (1995) Mathematical Statistics and Data Analysis, 2nd Edition, Duxbury ress. 5