Hi, could someone help me out with this question please?

Let X be the life-time of a hardware. We know that X Expon (y_{1/8}) ,

where y_{1/8 }is the 1/8-percentile of the random variable Y given by the p.d.f. : f_{Y}(x) = 3x^{2}, for 0≤ x ≤ 1

Recall that for any random variable Y we have F(y_{p}) = p for any 0 < p < 1. and we have q = P( X 1 ).

The hardware is installed at the beginning of day 1 and is monitored only once at the beginning of each day. Compute the probability P(X n) for every positive integer n, i.e. the probability that our hardware is functional for at least n days.

How is the memroyless property revealed here?

Let Z be the random variable which describes the number of days the hardware has been functional before failing, e.g if at the first observation one day after the installation (namely at the beginning of day 2) the hardware does not work then Z=1. Similarly if the hardware was functional during the first n - 1 monitorings but for n^{th}time it did not work then Z= n , for all n 1.

a) Find P(Z= k) , for k 0

b) is it true that Z id a geometric distribution with parameter p = 1 - q, with q defined above