Stat 5101 Lecture Notes - School of Statistics

5.3. BERNOULLI RANDOM VECTORS 151ThusE(X 1 | X 2 ) − M 12 M −122 X 2 = µ 1 − M 12 M −122 µ 2,which establishes the conditional expectation given in (5.33).To calculate the variance, we first observe thatvar(X 1 | X 2 )=W −111 (5.34)where W = M −1 is partitioned as in (5.32), because the quadratic form in theexponent of the density has quadratic term x 1 W 11 x 1 and Theorem 5.10 saysthat is the inverse variance matrix of the vector in question, which in this caseis x 1 given x 2 . We don’t know what the form of W 11 or it’s inverse it, but wedo know it is a constant matrix, which is all we need. The rest of the job canbe done by the vector version of the iterated variance formula (Theorem 3.7)var(X 1 )=var{E(X 1 |X 2 )}+E{var(X 1 | X 2 )} (5.35)(which we haven’t actually proved but is proved in exactly the same way as thescalar formula). We knowvar(X 1 )=M 11butvar{E(X 1 | X 2 )} + E{var(X 1 | X 2 )}=var{µ 1 +M 12 M −122 (X 2 − µ 2 )} + E{W11 −1 }= var(M 12 M −122 X 2)+W11−1= M 12 M −122 var(X 2)M −122 M′ 12 + W11−1= M 12 M −122 M 22M −122 M 21 + W11−1= M 12 M −122 M 21 + W11−1Equating the two givesM 11 = M 12 M −122 M 21 + W11−1which along with (5.34) establishes the conditional variance given in (5.33).5.3 Bernoulli Random VectorsTo start we generalize the notion of a Bernoulli random variables. One mightthink that should be a vector with i. i. d. Bernoulli components, but somethingquite different is in order. A (univariate) Bernoulli random variable is reallyan indicator function. All zero-or-one valued random variables are indicatorfunctions: they indicate the set on which they are one. How do we generalizethe notion of an indicator function to the multivariate case? We consider avector of indicator functions.We give three closely related definitions.