Some moments are E X i = np i, VarX i = np i (1 − p i), and CovX i Xj = − np i p j The multinomial distribution generalizes the binomial Consider an experiment having a total of r possible outcomes, and let the corresponding probabilities be p 1, , p r, respectively Now perform n independent replications of the experiment and let Xi record the total number of times that the iR For example, X2 Xevaluates to 0 on Z 2,the field of integers modulo 2,since 11 = 0 mod 2 We will say more about evaluation maps in Section 25,when we study polynomial rings 6 If Ris a ring,thenRX,the set offormalpowerseries a 0 a 1X a 2X 2 ··· with coefficients in R,is also a ring under ordinary addition and multiplicationGives a unique quotient Uniqueness of r(x) follows from uniqueness of q(x) (3) Let X n be the n× nmatrix whose (i,j)entry is x i,j Show det(X n) is irreducible in Cx i,j 1≤i,j≤n SOLUTION Since the determinant is linear in rows, no monomial term of det(X n) contains a subfactor of the form x2 i,j or
