1. ## 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)^(k-1) 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")

2. is >/ supposed to be $\ge$ ?

I haven't done this one in years...

$P(Y>a)=p\sum_{k=a+1}^{\infty}(1-p)^{k-1}$

let w=k-(a+1), so the sum starts at zero

$=p\sum_{w=0}^{\infty}(1-p)^{w+a}=p(1-p)^a\sum_{w=0}^{\infty}(1-p)^w$

$=p(1-p)^a\biggl({1\over 1-(1-p)}\biggr)=(1-p)^a$

SO

$P(Y>a+b|Y>a)={P(Y>a+b)\over P(Y>a)}={(1-p)^{a+b}\over (1-p)^a}$

$=(1-p)^b=P(Y>b)$

At least my memory for this memoryless property is decent.