1.4.8 Let Z be a random variable with the geometric probability mass function p.k/ D .1????/k; k D 0;1; : : : ;

where 0 <  < 1.

(a) Show that Z has a constant failure rate in the sense that PrfZ D kjZ  kg D 1???? for k D 0;1; : : : .

(b) Suppose Z0 is a discrete random variable whose possible values are 0;1; : : : , and for which PrfZ0 D kjZ0  kg D 1???? for k D 0;1; : : : . Show that the probability mass function for Z0 is p.k/.