This is a basic question, I am however not able to reduce my solution to the simple form proposed in a book and thus seeking help.
A patient can go via two stages in a disease. first stage is followed up by second stage and that is followed up by death.
Notation:
$p({t^{a}_{1})}$ be the pdf of time spent in stage A/stage 1
$p({t^{b}_{2})}$ be the pdf of time spent in stage B / stage 2 before death and this is independent of time spent in stage A
let $p(t^{d})$ be the convolution of $p(t^{a})$ and $p(t^{b})$ that gives probability distribution of months from start of disease till death i.e $p(t^{a}_{1}+t^{b}_{2})$.
From here on notation P refers to probability (& not pdf):
I further calculated probability of atleast total 'T' months (a constant) till death given stage 1 already happened should be $P(\frac{t>T}{t_{1}<T})= \frac{P(t^{d}>T) {\cap}P(t^{a}_{1} <= T)}{P(t^{a}_{1} <= T)}$.
I am however not able to tie this out to solution that's given out for
$P(\frac{t>T}{t_{1}<T}) = P(t^{a}_{1}<=T)-P(t^{d}<=T)$
Please can you suggest what I am doing incorrect here and help me derive?
