2.3 Show that Xn xa n x px(1 p)nx 1 B (a, n a 1) p 0 ya1(1

2.3 Show that Xn x¼a n

x

 

px(1  p)nx ¼

1 B

(a, n  a þ 1)

ðp 0

ya1(1  y)na dy for any 0

(a, naþ1). Thus, if X is a binomial random variable with parameters n and p, the probability that X is less than or equal to a (a¼0, 1, . . . , n) is 1  Ip(a þ 1, n 

a) ¼ I1p(n 

a, a þ 1)