I'm trying to interpret the following situation. In an economy, let $T$ denote the remaining lifetime (a stochastic variable) with exponential distribution and a Cumulative distribution function satisfying the following differential equation:
$$F'(t) = (1- F(t))p$$.
I would like to interpret this equation and the parameter p.
My attempt is to observe that the instantaneous rate of change of the cumulative distribution; i.e., how rapidly the probability of observing $T≤t$ is increasing, is $F'(t)$. With this, I think that this rate is iqual to the probability $P(T>t)$ weighted by $p$. But this is not convincing.
And how about $p$?
$F'(t)=(1-F(t))p\Rightarrow \dfrac{F'(t)}{1-F(t)}=p \equiv \lim_{\Delta t\to\ 0} \dfrac{\mathbb{P}\{T \in [t,t+\Delta t)|T \geq t\}}{\Delta t}=p $
so the numerator of this expression is the conditional probability that the event will occur in the interval $[t,t+\Delta t)$ given that it has not occurred before, and the denominator is the width of the interval. Dividing one by the other we obtain a rate of event occurrence per unit of time. Taking the limit as the width of the interval goes down to zero, we obtain an instantaneous rate of occurrence, which is $p$.