Renewal Models

The lifetime T (in months) of batteries used to power a personal stereo has survivor function
Q(t) = (8t2 + 4t +1)e-4t, t>=0.

(i) What proportion of batteries last for more than one month?

(ii) Find the density function g(t) of T, identify the distribution of T. Hence or otherwise find the mean and standard deviation of T.


After the personal stereo has been used for a long time (with batteries replaced when necessary) it is dropped and broken. The battery in it at the time is removed and placed in a similar personal stereo.

(iii) Show that the distribution of the total lifetime of the battery in use when the personal stereo is broken is Gamma(4,4). Hence write down the mean total lifetime of the such a battery.

(iv) Explain why the mean total lifetime of the battery removed from the broken stereo is no longer than the mean lifetime of a battery chosen at random and used in the new personal stereo.

(v) If the new personal stereo is kept for 2 years, how many times should you expect the battery to be replaced in that time?