covariance, expected value: i-th outcome has prob. p_i, with

[Ha_nguyen]

Please, help me with the following.

Suppose that an experiment can result in one of r possible outcomes, the i-[sup:31bx0r9l]th[/sup:31bx0r9l] outcome having probability p[sub:31bx0r9l]i[/sub:31bx0r9l] with i=1,2..,r and (sum from i=1 to r of p[sub:31bx0r9l]i[/sub:31bx0r9l]) =1.

Suppose n of these experiment are performed, and if the outcome of any of the n experiments does not affect the outcome of the n- 1 experiments.

Let Ni denote the number of times outcome i occur.
Find Cov(N[sub:31bx0r9l]i[/sub:31bx0r9l],N[sub:31bx0r9l]j[/sub:31bx0r9l] )
en compute the expected number of outcomes that do not occur

I have computed E(N[sub:31bx0r9l]i[/sub:31bx0r9l]) = npi
Var (N[sub:31bx0r9l]i[/sub:31bx0r9l]) = np[sub:31bx0r9l]i[/sub:31bx0r9l] (1-p[sub:31bx0r9l]i[/sub:31bx0r9l])

I also tried the Cov((N[sub:31bx0r9l]i[/sub:31bx0r9l],N[sub:31bx0r9l]j[/sub:31bx0r9l] ) = E(N[sub:31bx0r9l]i[/sub:31bx0r9l]N[sub:31bx0r9l]j[/sub:31bx0r9l] )-E((N[sub:31bx0r9l]i[/sub:31bx0r9l])E(N[sub:31bx0r9l]j[/sub:31bx0r9l] ) (but i dunno how to compute the first term,)

I got the hint from the teacher that N[sub:31bx0r9l]1[/sub:31bx0r9l]+N[sub:31bx0r9l]2[/sub:31bx0r9l]+...+N[sub:31bx0r9l]r[/sub:31bx0r9l] = n (a constant)
so I think Var (N[sub:31bx0r9l]1[/sub:31bx0r9l]+ N[sub:31bx0r9l]2[/sub:31bx0r9l]+..+N[sub:31bx0r9l]r[/sub:31bx0r9l]) = 0
but I dunno what to do next