Show that when k 2 the multinomial distribution reduces to

Show that when k = 2, the multinomial distribution reduces to the binomial distribution.

Solution

In multinomial distrbution,

f(x1...xk; n, p1,...pk) = [n!/(x1!...xk!)] p1^x1...pk^xk

If k = 2, then the subscripts only take values 1 and 2:

f(x1, x2; n, p1, p2) = [n!/(x1! x2!)] p1^x1 p2^x2

As x1 + x2 = n, then x2 = n - x1.

Also, p1 + p2 = 1. Then, p2 = 1 - p1.

Thus, everything can be expressed in terms of x1, p1, and n,

f(x1, x2; n, p1, p2) = [n!/(x1! (n-x1)!)] p1^x1 (1-p1)^(n-x1)

As we can see, dropping the subscripts, this is already a binomial distribution,

f(x, n, p) = [n!/(x! (n-x)!)] p^x (1-p)^(n-x) [DONE!]