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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 probability density function of time spent in stage A/ stage 1

$p({t^{b}_{2})}$ be the probability density function of time spent in stage B / stage 2 before demise and this is independent of time spent in stage A

let $p(t^{d}_{1+2})$ 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 $p(t^{a}_{1}+t^{b}_{2})$.

I further calculated probability of atleast total 'T' months till death given stage 1 already happened should be $P^{(b/a)}(t>T)= \frac{P(t^{d}>T) {\cap}P(t^{a}_{1} <= T)}{P(t^{a}_{1} <= T)}$.

I am however not able to tie this out to solution that's given out for

$P^{(b/a)}(t>T) = P(t^{a}_{1}<=T)-P(t^{d}<=T)$

Please can you suggest what I am doing incorrect here and help me derive?

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?

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 demise.

Notation:

$p({t^{a}_{1})}$ be the probability density function of time spent in stage A/ stage 1

$p({t^{b}_{2})}$ be the probability density function of time spent in stage B / stage 2 before demise and this is independent of time spent in stage A

let $p(t^{d}_{1+2})$ 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 $p(t^{a}_{1}+t^{b}_{2})$.

I further calculated probability density function of months till death given stage 1 already happened should be $p^{(b/a)}(t=T)=p(t^{d}=T)/P(t^{a}_{1} <= T)$. Is this correct?

This should mean that survival probability in stage 2 given stage 1 should be: $p_{survival}^{(b/a)}(t=T)=P(t^{d}>T)/P(t^{a}_{1} <= T)$

I am however not able to tie this out to solution that's given out to be

$P(t^{a}_{1}<=T)-P(t^{d}_{1+2}<=T)$

Please can you suggest what I am doing incorrect here?

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 probability density function of time spent in stage A/ stage 1

$p({t^{b}_{2})}$ be the probability density function of time spent in stage B / stage 2 before demise and this is independent of time spent in stage A

let $p(t^{d}_{1+2})$ 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 $p(t^{a}_{1}+t^{b}_{2})$.

I further calculated probability of atleast total 'T' months till death given stage 1 already happened should be $P^{(b/a)}(t>T)= \frac{P(t^{d}>T) {\cap}P(t^{a}_{1} <= T)}{P(t^{a}_{1} <= T)}$.

I am however not able to tie this out to solution that's given out for

$P^{(b/a)}(t>T) = P(t^{a}_{1}<=T)-P(t^{d}<=T)$

Please can you suggest what I am doing incorrect here and help me derive?

This is a basic question and I may not contribute much to understanding of other forum users. 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 demise.

Notation:

$p({t^{a}_{1})}$ be the probability density function of time spent in stage A/ stage 1

$p({t^{b}_{2})}$ be the probability density function of time spent in stage B / stage 2 before demise and this is independent of time spent in stage A

let $p(t^{d}_{1+2})$ 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 $p(t^{a}_{1}+t^{b}_{2})$.

I further calculated probability density function of months till death given stage 1 already happened should be $p^{(b/a)}(t=T)=p(t^{d}=T)/P(t^{a}_{1} <= T)$. Is this correct?

This should mean that survival probability in stage 2 given stage 1 should be: $p_{survival}^{(b/a)}(t=T)=P(t^{d}>T)/P(t^{a}_{1} <= T)$

I am however not able to tie this out to solution that's given out to be

$P(t^{a}_{1}<=T)-P(t^{d}_{1+2}<=T)$

Please can you suggest what I am doing incorrect here?

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 demise.

Notation:

$p({t^{a}_{1})}$ be the probability density function of time spent in stage A/ stage 1

$p({t^{b}_{2})}$ be the probability density function of time spent in stage B / stage 2 before demise and this is independent of time spent in stage A

let $p(t^{d}_{1+2})$ 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 $p(t^{a}_{1}+t^{b}_{2})$.

I further calculated probability density function of months till death given stage 1 already happened should be $p^{(b/a)}(t=T)=p(t^{d}=T)/P(t^{a}_{1} <= T)$. Is this correct?

This should mean that survival probability in stage 2 given stage 1 should be: $p_{survival}^{(b/a)}(t=T)=P(t^{d}>T)/P(t^{a}_{1} <= T)$

I am however not able to tie this out to solution that's given out to be

$P(t^{a}_{1}<=T)-P(t^{d}_{1+2}<=T)$

Please can you suggest what I am doing incorrect here?