Show, for the multinomial distribution with index m = 1, that the moments are i =

Show, for the multinomial distribution with index m = 1, that the moments are κ

i = πi

,

κ

ij = πiδij

, κ

ijk = πiδijk

¯¯¯¯¯¯¯and so on, where no summation is implied. Hence give an

κ

i = mπi

κ

i,j = m {πiδij − πiπj}

κ

i,j,k = m {πiδijk − πiπjδik [3] + 2πiπjπk}

κ

i,j,k,l = m{πiδijkl − πiπj (δikδjl [3] + δjkl [4]) + 2πiπjπkδil [6]

− 6πiπjπkπl}, alternative derivation of the first four cumulants in Exercise 2.16.