35. Using the 2-variable joint p.d.f. derived for the multinomial distribution and

(7.50), show that for any two components with i0 j: Cov½Ni;Nj ¼ npipj. (Hint:

First justify:

E½N1N2 ¼

Xn1 n1¼1 Xnn1 n2¼1 n1n2 n!pn1 1 pn2 2 ð1  p1  p2Þnn1n2 n1!n2!ðn  n1  n2Þ!

:

Then split this summation as the product Xn1 n1¼1 n1 n!pn1 1 ð1  p1Þnn1 n1!ðn  n1Þ! 

Xnn1 n2¼1 n2 ðn  n1Þ!

n2!ðn  n1  n2Þ!

p2 1  p1 n2 1  p1  p2 1  p1 nn1n2

;

and note that this second summation is E½n2 with a certain binomial distribution.

Alternatively, start with the double summation above, simplify n1n2 n1!n2! , and look for the binomial theorem.)