
Discrete
I'm doin revision questions and I came along this question:
CAn anyone help ?
1. Suppose that the discrete random variable X has a geometric distribution with parameter p (0 < p < 1). In other words, suppose that
P(X = k) = (1  p)^(k1) p for k = 1, 2..
Let Y = X  1.
(a) Find P(Y >/ l) for each l = 0,1,2,
(b) Show that for all nonnegative integers s and t
P(Y >/ s + t Y >/ s) = P(Y >/ t)
(c) Suppose that you are working in a call centre and let Y be length of time in seconds
that it takes you to answer a customer query.
(i) Describe the event {Y >/ s} in words.
(ii) Describe the result stated in (b) in words.
(Note that the result stated in (b) is called the "memoryless property")

is >/ supposed to be $\displaystyle \ge$ ?
I haven't done this one in years...
$\displaystyle P(Y>a)=p\sum_{k=a+1}^{\infty}(1p)^{k1}$
let w=k(a+1), so the sum starts at zero
$\displaystyle =p\sum_{w=0}^{\infty}(1p)^{w+a}=p(1p)^a\sum_{w=0}^{\infty}(1p)^w$
$\displaystyle =p(1p)^a\biggl({1\over 1(1p)}\biggr)=(1p)^a$
SO
$\displaystyle P(Y>a+bY>a)={P(Y>a+b)\over P(Y>a)}={(1p)^{a+b}\over (1p)^a}$
$\displaystyle =(1p)^b=P(Y>b)$
At least my memory for this memoryless property is decent.