I'm having some trouble understanding the mechanics of how to solve with this distribution. The question:
The number of years that a washing machine functions is a random variable whose hazard rate function is given by
$$
\lambda(t)=
\begin{cases}
0.2, & 0 < t < 2 \\
0.2 + 3(t-2), & 2 \leq t \leq 5 \\
1.1, & t > 5 \\
\end{cases}
$$
(a) What is the probability that the machine will still be working $6$ years after being purchased?
(b) If it is still working $6$ years after being purchased, then what is the conditional probability that it will fail within the next $2$ years?
I understand the formula is $λe^{-λt}$ and I think I need to integrate this for $t$ from $0$ to $6$.
However my answer is way off from the answer in the solutions page.
Does anybody have time for a step by step solution?
Much appreciated.
Best Answer
$\displaystyle P\{X > T\} =\exp\left(-\int_0^T h(t)\,\,\mathrm dt\right)$ where $h(t)$ is the hazard rate. So calculate the area under the hazard rate function from $0$ to $6$ and apply the formula to get the probability that the machine has not failed in the first $6$ years.
For part (b), you are asked for $\displaystyle P\{X \leq 8\mid X > 6\} = \frac{P\{\{X \leq 8\}\cap \{X >6\}\}}{P\{X > 6\}} = \frac{P\{6 < X \leq 8\}}{P\{X > 6\}}$ which you should be able to work out for yourself based on the relationship between the hazard rate and the complementary CDF that is given in the first paragraph.