By Jeremy D. Finn

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R denote a set of r-dimensional vectors such that for k = 1, 2, . . , r, the vector εk has a one in the k-th position and zeros elsewhere. If r = 3, for example, then this set of vectors would be ε1 = (1, 0, 0), ε2 = (0, 1, 0), and ε3 = (0, 0, 1). In what follows, these vectors will be used as indicators to record which one of the r events occurs at each independent trial. If, for example, event Ak occurs, then the occurrence of this event will be denoted by εk . A sequence of N independent observations will be denoted by a sequence of the form εi1 εi2 .

16) = k1 ! × k2 ! × . . × kr ! These numbers are also known as the multinomial coeﬃcients, because they appear in the multinomial theorem. Let aν for ν = 1, 2, . . , r for a set of real numbers. Then, the multinomial theorem is the statement or equation ∑ N! N (a1 + a2 + · · · + ar ) = ak1 ak2 × . . × akr r , k1 ! × k2 ! × . . × kr ! 18) such that kν ≥ 0 is a non-negative integer for ν = 1, 2, . . , r. The proof of this theorem is very similar to that for the binomial theorem and will, therefore, be omitted.

7) x x=0 To derive a formula for the variance of X, it will be necessary to ﬁnd a formula for this expectation. 4) with respect to s. Then, ( ) N d2 g (s) ∑ N −x N −2 x−1 N = x (x − 1) s px (1 − p) = N (N − 1) p2 (ps + q) . 8) By setting s = 1 in this equation, it can be seen that ( ) N ∑ N x N −x E [X (X − 1)] = x (x − 1) p (1 − p) = N (N − 1) p2 . 9) x x=0 However, [ ] E X 2 = E [X (X − 1)] + E [X] = N (N − 1) p2 + N p. 10) Therefore, var[X] = N (N − 1) p2 + N p − N 2 p2 = N p − N p2 = N p (1 − p) = N pq.