What is the delta t on the conditional probability of failure (TTF)

Question

Can someone explain me what is the $\Delta$t of the conditional probability of failure (CPF) on hazard function? CPF = P { t $\le$ T $\le$ t+$\Delta t$} $\mid$ T > t}

$$CPF = \frac{R(t) - R(t- \Delta t )}{R(t)}$$

Answer 1

$\Delta t$ is just "some (positive) increment on $t$".   This is usually a small increment.

$\mathsf P(t\leq T\leq t+\Delta t\mid T> t)$ is the probability that $T$ lies in some constant (positive) interval $\Delta t$ following $t$ given that $T$ is greater than $t$.

It's a convenient standard to understand $\Delta$*variable* to mean an increment of the variable; rather than introducing a new constant and needing to explaining what that is each and every time.   Though this notation should be explained at least once somewhere in the text, it is so standard some lazy authors may take it for granted that the reader will already understand.