What is the memoryless property?
Read 3.2 of this: Exponential distribution - Wikipedia, the free encyclopedia
let X be the lifetime of the washing machine
$\displaystyle f(x) = pe^{-px} $
$\displaystyle P(X>x+t|X>x) = \frac{P(X>x,X>x+t)}{P(X>x)} $
EDIT: $\displaystyle = \frac{P(X>x+t)}{P(X>x)} = \frac{e^{-p(x+t)}}{e^{-px}}$ $\displaystyle = \frac{e^{-px}e^{-pt}}{e^{-px}} = e^{-pt}$
notice that $\displaystyle P(X>t) = \int_{t}^{\infty} pe^{-px}dx = e^{-pt} $
so the probabilty that the washing machine will continue working for at least another t years is the same probability if the washing machine were new
I made two errors. But it doesn't change the final result.
$\displaystyle \frac{P(X>x+t)}{P(X>x)} = \frac{\int^{\infty}_{x+t} pe^{-px} \ dx} {\int^{\infty}_{x} pe^{-px} \ dx} $$\displaystyle = \frac{e^{-p(x+t)}}{e^{-px}} = \frac{e^{-px}e^{-pt}}{e^{-px}}= e^{-pt}= P(X>t)$
I get:
$\displaystyle 1+e^{-x^{p+t}}$ divided by $\displaystyle 1+e^{-x^{p}}$
Which then leads to
$\displaystyle e^0 +e^{-x^{p+t}}$ divided by $\displaystyle e^0 +e^{-x^p}$
Which equals
$\displaystyle e^{-x^{p+t-p}}$
=
$\displaystyle e^{-x^{p}}$
Almost certain I'm wrong,