Statistics Help

Question 1 [20 marks]. This question demonstrates a special feature of the
geometric distribution that makes it suitable for applications to discrete waiting times.
It is called the memoryless property. This is a characteristic of the geometric
random variable that might or might not correspond to what happens in real-life
applications.
Assume that X models the first day that a server will fail. On each day the server fails
with probability p, independently of the other days. Equivalently, X ∼ Geom(p).
(a) For a positive integer, n1, using the known expression for the p.m.f. of X, prove
that
P(X > n1) = (1 − p)
n1
.
Deduce an expression for the c.d.f. P(X ≤ n1).
Hint: Use the formula for the sum of a geometric series 1 + r + r
2
· · · for suitable
r.