15. For the geometric distribution, let m0m 1E jm p Py j0 jm1 p j...

15. For the geometric distribution, let m0m 1E½ jm ¼ p Py j¼0 jmð1  pÞ j for m A N and mb1, and analogously, m00

¼ 1.

(a) Show that these moments can be produced iteratively by m0m ¼

1  p p

Xm1 j¼0 m

j m0j :

(Hint: Show that for mb1, Py j¼1 jmð1  pÞ j ¼ ð1  pÞ½1 þ

Py j¼1ð j þ 1Þmð1  pÞ j , and use the binomial theorem.)

(b) Derive the mean and variance formulas in (7.103) from part (a).